Cignoli,mundici-algebraic Foundations Of Many-valued Reasoning.pdf

ALGEBRAIC FOUNDATIONS OF MANY-VALUED REASONING

TRENDS IN LOGIC Studia Logica Library VOLUME7 Managing Editor Ryszard W6jcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Daniele Mundici, Department of Computer Sciences, University of Milan, Italy Graham Priest, Department of Philosophy, University of Queensland, Brisbane, Australia Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada Heinrich Wansing, Institute of Philosophy, Dresden University ofTechnology, Germany Assistant Editor Jacek Malinowski, Box 61, UPT 00-953, Warszawa 37, Poland

SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica - that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and SOUfces of inspiration is open and evolves over time.

The titles published in this series are listed at the end afthis valurne.

ROBERTO L.O. CIGNOLI Department of Mathematics. University of Buenos Aires. Argentina

ITALA M.L. D'OTfAVIANO Department of Philosophy and The Centre for Logic. Epistemology and the History of Science. State University of Campinas. Brazil

and

DANIELE MUNDICI Department of Computer Science. University of Milan. Italy

ALGEBRAIC FOUNDATIONS OF MANY-VALUED REASONING

Springer-Science+Business Media, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5336-7 ISBN 978-94-015-9480-6 (eBook) DOI 10.1007/978-94-015-9480-6

Printed on acid-free paper

All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000. Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, incIuding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

To the memory

0/

ROLANDO CHUAQUI ENNIO DE GlORGI ANTONIO MONTEIRO

great Scientists and Teachers

Contents Introduction

1

1 Basic not ions 1.1 MV-algebras . . . . . . . . . . . . 1.2 Homomorphisms and ideals '" 1.3 Subdirect representation theorem 1.4 MV-equations . . 1.5 Boolean algebras . . . . 1.6 MV-chains . . . . . . . . 1.7 Bibliographical remarks

7 7 12 19

20 24 27

29

2 Chang completeness theorem 2.1 The functor r . . . . . . . . 2.2 Good sequences . . . . . . . 2.3 The partially ordered monoid M A 2.4 Chang's f-group GA' . . . . . 2.5 Chang completeness theorem. 2.6 Bibliographical remarks

31

3 Free MV-algebras 3.1 McN aughton functions . . . . . . . . 3.2 The one-dimensional case . . . . . . . 3.3 Decomposing McNaughton functions 3.4 Ideals in free MV-algebras 3.5 Simple MV-algebras .. 3.6 Semisimple MV-algebras 3.7 Bibliographical remarks

51

vii

31 34

37

40 43 49

51 56 62

64 70 72 75

viii

CONTENTS

4 Lukasiewicz oo-valued calculus 4.1 Many-valued propositional calculi 4.2 Wajsberg algebras. . 4.3 Provability.......... 4.4 Lindenbaum algebra . . . . 4.5 All tautologies are provable 4.6 Syntactic and semantic consequence . 4.7 Bibliographical remarks . . . . . . .

· 101

5 Ulam's game 5.1 Questions and answers . . . . . . . 5.2 Dynamics of states of knowledge .. 5.3 Operations on states of knowledge . 5.4 Bibliographical remarks ..

· 103 .104 · 107 · 109

6 Lattice-theoretical properties 6.1 Minimal prime ideals . . . . . . . . . . . . 6.2 Stonean ideals and archimedean elements . 6.3 Hyperarchimedean algebras . . . 6.4 Direct products . . . . . . . . . . 6.5 Boolean products of MV-algebras 6.6 Completeness . . . . . . . . . . 6.7 Atoms and Pseudocomplements 6.8 Complete distributivity . 6.9 Bibliographical remarks 7 MV-algebras and f-groups 7.1 Inverting the functor r 7.2 Applications . . . . . 7.3 The radical . . . . . . 7.4 Perfeet MV-algebras . 7.5 Bibliographical remarks 8 Varieties of MV-algebras 8.1 Basic definitions. . . . . 8.2 Varieties from simple algebras 8.3 MV-chains of finite rank . . .

77

78 82

87 92 94

97 103

111 .112 .115 .116 · 121 · 124 · 129 · 132 · 134 · 137

139 · · · · ·

139 146 150 151 156

157 · 157 · 160 · 161

CONTENTS 8.4 8.5 8.6 8.7

Komori's c1assification . . . . . . . . Varieties generated by a finite chain . The cardinality of Free~ Bibliographical remarks . . . . . . .

ix · 167 · 171 .173 · 177

9 Advanced topics 9.1 McNaughton's theorem . . . . . . . . 9.2 Nonsingular fans and normal forms . 9.3 Complexity of the tautology problem 9.4 MV-algebras and AF C*-algebras 9.5 Di Nola's representation theorem 9.6 Bibliographical remarks . . . . .

. . . . . .

179 180 185 187 191 193 194

10 Further Readings 10.1 More than two truth values 10.2 Current Research Topics . . 10.2.1 Product . . . . . . . 10.2.2 States, observables , Probability, Partitions . 10.2.3 Deduction . . . . . . . 10.2.4 Further constructions . . . . . . . . . . . ..

197 . 197 . 199 . 199 . 200 . 201 . 201

Bibliography

203

Index

225

Introduction The aim of this book is to give self-contained proofs of all basic results concerning the infinite-valued proposition al calculus of Lukasiewicz and its algebras, Chang's MV-algebras. This book is for self-study: with the possible exception of Chapter 9 on advanced topics, the only prerequisite for the reader is some acquaintance with classical propositional logic, and elementary algebra and topology. In this book it is not our aim to give an account of Lukasiewicz's motivations for adding new truth values: readers interested in this topic will find appropriate references in Chapter 10. Also, we shall not explain why Lukasiewicz infinite-valued propositionallogic is a basic ingredient of any logical treatment of imprecise notions: Hajek's book in this series on Trends in Logic contains the most authoritative explanations. However, in order to show that MV-algebras stand to infinite-valued logic as boolean algebras stand to two-valued logic, we shall devote Chapter 5 to Ulam's game of Twenty Questions with lies/errors, as a natural context where infinite-valued propositions, connectives and inferences are used. While several other semantics for infinite-valued logic are known in the literature-notably Giles' gametheoretic semantics based on subjective probabilities-still the transition from two-valued to many-valued propositonallogic can hardly be modelled by anything simpler than the transformation of the familiar game of Twenty Questions into Ulam game with lies/errors. This book is mainly addressed to computer scientists and mathematicians wishing to get acquainted with a compact body of beautiful results and methodologies-that have found applications in the treatment of uncertain information, (e.g., adaptive error-correcting codes) as weIl as in various mathematical areas, such as toric varieties, lattice1 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

2

INTRODUCTION

ordered groups and C"-algebras. As the title indicates, the main emphasis is on algebraic methods. Thus, reversing the historical order, we shall make the reader familiar with MV-algebras before introducing Lukasiewicz's propositional calculus in Chapter 4. This will allow us to get neat and elementary proofs of several deep results, using much less symbolism and detail than in traditional syntax-oriented approaches. The definition-theorem-proof style adopted throughout this book will hopefully result in time saving for the reader who wishes to get the proofs of all main theorems on the infinite-valued calculus as quickly as possible, without embarking on a potentially unbounded search through a scattered literature on ordered groups, lattices, algebraic logic, polyhedra, geometry of numbers, model theory, linear inequalities, et cetera. By definition, an MV-algebra A is a set equipped with an associative-commutative operation EB, with a neutral element 0, and with an operation -, such that -,-,x = x, x EB -,0 = -,0, and, characteristically,

These six equations are intended to capture some properties of the real unit interval [0, 1] equipped with negation -,x = 1 - x and truncated addition x EB y = min(l, x + y). For instance, once interpreted in [0,1], the left hand term in the last equation coincides with the maximum of x and y; thus the equation states that the max operation over [0,1] is commutative. The fundamental theorem on MV-algebras is Chang's completeness theorem, stating that every valid equation in [0, 1] is automatically valid in all MV-algebras. A new proof of this theorem is given in Chapter 2. As a preliminary step, in Chapter 1 we prove Chang's subdirect representation theorem, stating that an equation is valid in every MY-algebra iff it is valid in every totally ordered MV-algebra. As in the classical case, one may ask for an effective procedure to decide when an equation is valid. Rather than working in "MYalgebraic equationallogic", it is more convenient to give the Lukasiewicz infinite-valued calculus the same role that the classical propositional calculus has for the boolean decision problem. Accordingly, one may write -,x ~ y instead of x EB y

INTRODUCTION

3

and transform valid MV-equations into tautologies by writing p ~ q instead of p = q. The main theorem proved in Chapter 4 then states that the rule of modus ponens is sufficient to obtain all tautologies in the infinite-valued calculus of Lukasiewicz starting from four basic tautologies (originally given by Lukasiewicz) corresponding to the above defining equations for MV-algebras. These equations are thus "complete", in the sense that every equation that is valid for [0, 1] is obtainable from them by substituting equals for equals. As another corollary, in Chapter 5 we shall show that tautologies in the infinitevalued calculus of Lukasiewicz coincide with those formulas that are true in every Ulam game, independently of the number of errors/lies. Having thus acquired a unified view of valid MV-equations and infinite-valued tautologies, we can handle logical notions using standard algebraic methods. Thus, e.g., from the logic-algorithmic viewpoint, free MV-algebras over n generators consist of all equivalence classes f of formulas in n variables. On the other hand, from the algebraic-geometric viewpoint, free MV-algebras consist of all continuous [0, IJ-valued piecewise linear functions f with integer coefficients defined over the cube [0, 1Jn. (This is McNaughton's representation theorem. See Chapter 3 for the case of one variable, and Chapter 9 for the general case). The multiple nature of f is useful in the study of normal form reductions, and it yields a concrete visualization of the not ion of consequence and its fine structure. Further , one can strengthen the completeness theorem and give an algorithm to decide whether a formula is a tautology-having no greater complexity than its analog for boolean tautologies (see Chapter

9). Generalizing the relationship between the interval [0, 1] and the naturally ordered additive group of real numbers, every MV-algebra A can be realized as the unit interval A = [0, u] = r(G, u) of a unique abelian lattice-ordered group G with a strong unit u equipped with negation u - x and truncated addition x EB y = (x + y) /\ u. Specifically, as proved in Chapter 7, r is a categorical equivalence between abelian lattice-ordered groups with strong unit, and MV-algebras. Among the many important consequences of this equivalence, one can unambiguously say, e.g., that elements al, ... , an E A "sum up to one", or that they are " linearly independent". These two conditions allow one to

4

INTRODUCTION

give an MV-algebraic definition of "partition of unit", thus generalizing the basic notion of boolean partition. Since abelian lattice-ordered groups are so weH established-their roots going back to the time-honored theory of magnitudes-one might wonder, why MV-algebras should be given special attention. One main reason is that, while a strong unit u in a group G of magnitudes is no less important than the zero element, the property of u being a strong unit in Gis formalized by for aH x E G there is n = 0, 1,2, ... such that x::; nu. This archimedean-like property is not only beyond the expressive power of equations, but, by Gödel's incompleteness theorem, is undefinable even in first-order logic. Remarkably enough, up to categorical equivalence, lattice-ordered abelian groups with a strong unit can be defined by equations-the equations of MV-algebras. Since these equations are not hing more than a reformulation of tautologies in the infinite-valued calculus, and since the infinite-valued tautology problem is no more complicated than its boolean counterpart, it becomes natural to apply to other mathematical areas logic-algorithmic not ions originating from the many-valued calculus. For this purpose, one may use lattice-ordered abelian groups with strong unit, via the r functor, as a bridge between MV-algebras and other structures. Important examples are given by approximately finite-dimensional (AF) O'-algebras, the algebras of operators currently used for the mathematical description of infinite spin systems. As another interesting example, disjunctive normal form reductions for formulas in the infinite-valued calculus are essentiaHy the same as desingularization algorithms for toric varieties, once the latter are described by their associated fans-a fan being a complex of rational polyhedral cones. The relationships between MV-algebras, O'-algebras, and toric varieties will be briefly discussed in Chapter 9; most of that chapter can be safely skipped by readers only interested in the Lukasiewicz calculus and its algebras. Similarly, we have made no attempt to introduce here "first-order" infinite-valued logic: as a matter of fact, the development of the infinite-valued counterparts of such notions as "set",

INTRODUCTION

5

"equality", "structure" would result in a much less elementary textbook of considerably larger size-Iet alone the problem of choosing the right definitions. In a final section, we give appropriate bibliographical references to the interested reader. While it would be beyond the scope of the book to cover all the fascinating and rapidly developing fields of research connected with infinitevalued reasoning, for most of these topics the book is intended to provide sufficient background material. Thus, Chapter 8 is devoted to the classification of equational classes of MV-algebras; these include all classes associated to the finite-valued calculi of Lukasiewicz. The fruitful interplay between MV-algebras and lattices is discussed at length in Chapter 6. Extensive bibliographical references are given in the final chapter on Further Readings, concerning such basic issues as the "multiplication connective", as weIl as probability and proof theory in the infinite-valued propositional calculus of Lukasiewicz.

*** While the first germ of this book is our monograph [58], the proofs of many fundamental theorems are given here in a more general and self-contained form, using results that have appeared in the literature after the publication of [58]. Several sections on advanced topics have been added, and the bibliography has been considerably expanded. The book also contains a wealth of previously unpublished material. This book is didactic in its spirit: preliminary versions have been tested in several graduate courses in Bahfa Blanca, Barcelona, Buenos Aires, Campinas, Merida, Milan, Patras, and in the general context of the European Project known as Action COST number 15 on "Manyvalued Logic for Computer Science Applications". We are grateful to Stefano Aguzzoli, Agata Ciabattoni, Vincenzo Marra and Claudia Picardi for their valuable comments on earlier drafts of this book. For valuable hints and discussions we are indebted to many more students and colleagues: we ask them to forgive us for not listing them here. We also thank the anonymous referee for his careful and competent reading and for suggesting several improvements. We gratefully acknowledge partial support from the National Research Councils of Argentina (CONICET), Brazil (CNPq) and Italy

6

INTRODUCTION

(CNR), as weH as from "Fundaci6n Antorchas" (Buenos Aires), "Centre de Recerca Matematica, Institut d'Estudis Catalans" (Barcelona), Funda<;äo de Amparo a Pesquisa do Estado de Säo Paulo (FAPESP, Säo Paulo), Istituto Italiano per gli Studi Filosofici (NapIes), besides our own universities, and COST Action 15. Their support allowed us to meet several times in Argentina, Brazil and Italy. R.C., LM.L.D.O, D.M.

Chapter 1 Basic nations We introduce MV-algebras by means of a small number of simple equations, in an attempt to capture certain properties of the unit real interval [0,1] equipped with truncated addition x tf) y = min(1, x + y) and negation 1 - x. We show that every MV-algebra contains a natural lattice-order. The chapter culminates with Chang's Subdirect Representation Theorem, stating that if an equation holds in all totally ordered MV-algebras, then the equation holds in all MV-algebras.

1.1

MV-algebras

Definition 1.1.1 An MV-algebra is an algebra (A, tf), -', 0) with a binary operation tf), a unary operation -, and a constant 0 satisfying the following equations:

MV1) x

tf)

(y tf) z) = (x

MV2) x

tf)

y

MV3) x

tf)

0= x

tf)

y)

tf)

z

= y tf) x

MV4) -,-,x = x MVS) x

tf)

-.0 = -,0

MV6) -,( -,x tf) y)

tf)

y

= -,( -,y tf) x)

tf)

x.

7 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

8

CHAPTER 1. BASIC NaTIONS

In particular, axioms MV1)-MV3) state that (A, EB, 0) is an abelian monoid. Following tradition, we denote an MV-algebra (A, EB, -', 0) by its universe A. A singleton {O} is a trivial example of an MV-algebra. An MValgebra is said nontrivial Hf its universe has more than one element. As a first example of a nontrivial MV-algebra, consider the real unit interval [0,1] = {x E RIO ~ x ~ I}, and for all x,y E [0,1], let x EB y =def min(l, x + y) and -,x =def 1 - x. It is easy to see that [0,1] =def ([0,1], EB, -', 0) is an MV-algebra. As a second example, if (A, V, 1\, -,0,1) is a boolean algebra, then (A, V, -,0) is an MValgebra, where V, - and Odenote, respectively, the join, the complement and the smallest element in A. A subalgebra of an MV-algebra A is a subset S of A containing the zero element of A, closed under the operations of A, and equipped with the restriction to S of these operations. The intersection of any nonempty family of subalgebras of A is a subalgebra of A. Given a subset X of A the intersection of all sub algebras of A containing X is a subalgebra of A, and is called the subalgebra 0/ A generated by X. The rational numbers in [0,1], and, for each integer n ~ 2, the n-element set

Ln =def {O, l/(n - 1), ... , (n - 2)/(n - 1), I}, yield examples of subalgebras of [0,1]. Given an MV-algebra A and a set X, the set A X of all functions /: X -+ A becomes an MV-algebra if the operations EB and -, and the element are defined pointwise. The continuous functions from [O,lJ into [0, 1J form a subalgebra of the MV-algebra [0, l]l°,lJ. On each MV-algebra A we define the constant 1 and the operations 8 and e as follows:

°

(1.1)

1 =def -,0,

(1.2)

X8Y=def-,(-,xEB-,y),

(1.3)

xe Y =def X 8 -'y.

An MV-algebra is nontrivial if and only if identities are immediate consequences of MV 4):

°i= 1.

The following

1.1. MV-ALGEBRAS

MV7) -,1

= 0,

9

and

MVS) x Ee y = -,(-,x 0 -,y). Axioms MV5) and MV6) can now be written as:

MV5' ) x Ee1 = 1, and MV6') (x9y)Eey= (y9x)Eex. Setting y = -,0 in MV6) we obtain:

MV9) x Ee -,x = 1. Note that in the MV-algebra [0,1] we have x0y and x 9 y = max(O, x - y).

= max(0,x+y-1)

Notation: Following common usage, we consider the -, operation more binding than any other operation, and the 0 operation more binding than Ee and 9. Lemma 1.1.2 Let A be an MV-algebra and x, y E A. lowing conditions are equivalent:

Then the lol-

(i) -,xEey=l;

(ii) x0-,y=0; (iii) y = x Ee (y 9 x); (iv ) there is an element z E A such that x Ee z = y. Proof: (i) =? (ii) By MV4) and MV7). (ii) =? (iii) Immediate from MV3) and MV6'). (iii) =? (iv) Take z = y 9 x. (iv) =? (i) By MV9), -,x EI1 x EI1 z = 1. 0

Let A be an MV-algebra. For any two elements x and y of A let us agree to write x~y

iff x and y satisfy the above equivalent conditions (i)-(iv). It follows that ~ is a partial order, called the natural order of A. Indeed, reflexivity

10

CHAPTER 1. BASIC NOTIONS

is equivalent to MV9), antisymmetry follows from conditions (ii) and (iii) , and transitivity follows from condition (iv). An MV-algebra whose natural order is total is called an MV-chain. Note that, by (iv), the natural order of the MV-chain [0,1] coincides with the natural order of the real numbers.

Lemma 1.1.3 Let A be an MV-algebra. For each a E A, -,a is the unique solution x of the simultaneous equations: (1.4) {aEBx = a0x =

1 0

Proof' By Lemma 1.1.2, these two equations amount to writing -,a ::; x ::; -,a. 0

Lemma 1.1.4 In every MV-algebra A the natural order::; has the following properties:

(i) x::; Y if and only if -,y ::; -,x;

(ii) If x::; y then for each z

E A, x EB z ::; y EB z

and x 0 z ::; y 0 z;

(iii) x0y::; z iJJx::; -'yEBz. Proof: (i) This follows from Lemma 1.1.2(i), since -,xEBy = -'-'yEB-,x. (ii) The monotonicity of EB is an easy consequence of Lemma 1.1.2(iv); using (i), one immediately proves the monotonicity of 0. (iii) It is sufficient to note that x 0 y ::; z is equivalent to 1 = -,(x 0 y) EB z = -,x EB -'y EB z. 0

Proposition 1.1.5 On each MV-algebra A the natural order determines a lattice structure. Specijically, the join x V y and the meet x 1\ y of the elements x and y are given by (1.5) x V y = (x 0 -,y) EB y = (x

e y) EB y,

=x0

(-,x EB y).

(1.6) x 1\ y = -,( -,x V -,y)

1.1. MV-ALGEBRAS

11

Proof: To prove (1.5), by MV6'), MV9) and Lemma 1. 1.4(ii) , x :::; (x e y) E9 y and y:::; (x e y) E9 y. Suppose x :::; z and y:::; z. By (i) and (iii) in Lemma 1.1.2, -,x E9 z = 1 and z = (z e y) E9 y. Then by MV6') we can write

-,((x e y) E9 y) E9 z = (-,(x e y)

e y) E9 y E9 (z e y)

= (ye -,(x e y)) E9 -,(x e y) E9 (z e y) = (y =

(y

e -,(x e y)) E9 -,x E9 y E9 (z e y) e -,(x e y)) E9 -,x E9 z = 1.

It follows that (xey) E9y:::; z, which completes the proof of (1.5). We now immediately obtain (1.6) as a consequence of (1.5) together with Lemma 1.1.4(i). 0 Proposition 1.1.6 The following equations hold in every MV-algebra:

(i) x0(yVz)=(x0y)V(x0z), (ii) x E9 (y!\ z) = (x E9 y) !\ (x E9 z). Proof: By MV6') and Lemma 1. 1.4(ii) , x 0 y :::; x 0 (y V z) and x 0 z :::; x 0 (y V z). Suppose x 0 y :::; t and x 0 z :::; t. Then by Lemma 1.1.4(iii), y:::; -,x E9 t and z :::; -,x ElH, whence y V z :::; -,x tJj t. One more application of Lemma 1.1.4 (iii) yields (y V z) 0 x :::; t, which completes the proof of (i). It is now easy to see that (ii) is a consequence of (i), using Lemma 1.1.4(i), together with MV4) and MV8). 0

Proposition 1.1. 7 Every MV-algebra satisfies the equation

(1.7) (xey)!\(yex)=o. Proof: By making repeated use of MV6) and its variants, together with the basic properties of the operations E9 and 0 we obtain:

(xey)!\(yex)

12

CHAPTER 1. BASIC NOTIONS

= (x e y) 8 (-,(x e y) E9 (y e x» = x 8 -,y 8 (y E9 -,x E9 (y e x» = x 8 (-,x E9 (y

e x»

0 (-,(-,x E9 (y e x» E9 -,y)

= (y e x) 8 (-,(y e x) E9 x) 8 (-,(-,x E9 (y e x» = y 8 -,x 8 (-,(y e x) E9 x) 8 ((x 0 -,(y e x»

E9 -,y)

E9 -,y)

= -,x 8 (x E9 -,(y e x» 8 y 0 (-,y E9 (x 8 (-,y E9 x») = -,x 8 (x E9 -,(y e x» 8 (x 0 (-,y E9 x» 8 (-,(x 8 (-,y E9 x» E9 y)

= 0,

since by MV8) and MV9), -,x 8x

= O.

0

Let A be an MV-algebra. For each x E A, we let Ox = 0, and für each integer n ~ 0, (n + l)x = nx E9 x. Lemma 1.1.8 Let x and y be elements of an MV-algebra A. x 1\ Y = 0 then for each integer n ~ 0, nx 1\ ny = O.

If

Proof: If x 1\ Y = 0 then by monotonicity (Lemma 1.1.4) and distributivity (Proposition 1.1.6), x = xE9(xl\y) = (xE9x) 1\ (xEBy) ~ 2xl\y, whence 0 = x 1\ Y ~ 2x 1\ y. It follows that 0 = 2x 1\ 2y = 4x 1\ 4y = 8x 1\ 8y = ... . The desired conc1usion now follows from nx 1\ ny :s 2n x 1\ 2n y = O. 0

1.2

Homomorphisms and ideals

Let A and B be MV-algebras. A function h: A ---. B is a homomorphism Hf it satisfies the follüwing conditions, for each x, y E A:

Hl) h(O) = 0, H2) h(x E9 y) = h(x) E9 h(y),

H3) h(-,x) = -,h(x).

1.2. HOMOMORPHISMS AND IDEALS

13

Following current usage, if h is one-one we shall equivalently say that h is an injective homomorphism, or an embedding. If the homomorphism h: A - t B is onto B we say that h is surjective. By an isomorphism we shall mean a surjective one-one homomorphism. We write A C::! B Hf there is an isomorphism from A onto B. The kernel of a homomorphism h: A - t B is the set K er(h)

=dej

h-1(0)

= {x E AI h(x) = O}.

Our next aim is to characterize kerneis of homomorphisms. An ideal of an MV-algebra A is a subset 1 of A satisfying the following conditions:

11) 0 E 1, 12) If xE 1, y E A and y

~

x then y E 1,

13) If x E 1 and y E 1 then x ffi y EI. The intersection of any family of ideals of A is still an ideal of A. For every subset W ~ A, the intersection of all ideals 1 ;2 W is said to be the ideal generated by W, and will be denoted (W). The proof of the next lemma is immediate, and will be omitted. Lemma 1.2.1 Let W be a subset of an MV-algebra A. If W = 0, then (W) = {O}. If W =1= 0, then (W)

= {x E A I x

~ Wl EB··· EB Wk, fOT some Wl,"" Wk E W}.

o In particular, for each element z of an MV-algebra A, the ideal (z) = ({ z }) is called the principal ideal generated by z, and we have

(1.8)

(z)

= {x

E

AI nz ?

x for some integer n ? O}.

Note that (0) = {O} and (1) = A. Further, for every ideal J of an MV-algebra A and each z E A we have

(1. 9)

(J U {z}) = {x E A Ix

~

nz ffi a, for some n E N and a E J}.

An ideal 1 of an MV-algebra Ais proper Hf 1 =1= A. We say that 1 is prime Hf it is proper and satisfies the following condition:

CHAPTER 1. BASIC NaTIONS

14

14) For each x and y in A, either (x

e y)

E I or (y

e x)

E I.

An ideal I of an MV-algebra A is caHed maximal Hf it is proper and no proper ideal of A strictly contains I, Le., for each ideal J =I- I, if I ~ J then J = A. The next proposition generalizes a weH known property of maximal ideals in boolean algebras.

Proposition 1.2.2 Por any proper ideal J of an MV-algebra A the following conditions are equivalent: (i) J is a maximal ideal of A; (ii) for each x E A, x

rt. J

iff -mx E J for some integer n 2:: 1.

Proof: (i) ----+ (ii): Suppose that J is a maximal ideal of A. If x rt. J, then ({x}UJ) = A, and by (1.9), 1 = nxEBa for some integer n 2:: 1 and a E J. Stated otherwise, -,nx::; a E J, whence by 12), -,nx E J. Conversely, if x E J, then nx E J for each integer n 2:: 1; since J is proper, -mx rt. J. (ii) ----+ (i): Let K =I- J be an ideal of A such that J ~ K. For every x E K \ J we must have -,nx E J for some integer n 2:: 1. Hence 1 = nx EB -,nx E K, and K = A. 0 We denote by I(A), P(A) and M(A) the sets of ideals, prime ideals and maximal ideals of A, respectively. In the next lemma we summarize, for furt her reference, some easy relations between ideals and kernels of homomorphisms.

Lemma 1.2.3 Let A, B be MV-algebras, and h: A phism. Then the following properties hold:

----+

B a homomor-

(i) Por each ideal J of B, the set h-1(J) =def {x E AI h(x) E J} is an ideal of A. Thus in particular, K er(h) E I(A); (ii) h(x) ::; h(y) iff xe y E Ker(h); (iii) h is injective iff Ker(h) = {O}; (iv) K er (h) =I- A iff B is nontrivial;

1.2. HOMOMORPHISMS AND IDEALS

15

(v) K er(h) E P(A) iJ] B is nontrivial and the image h(A), as a subalgebra of B, is an MV-chain. 0 Definition 1.2.4 The distance function d: A x A (1.10) d(x, y) =def (x

----t

A is defined by

e y) EB (y e x).

In the MV-algebra [0,1], d(x, y) = Ix - yl. In every boolean algebra the distanee funetion eoincides with the symmetrie differenee operation. Proposition 1.2.5 In every MV-algebra A we have:

(i) d(x,y) = 0 iJ] x = y, (ii) d(x, y) = d(y, x), (iii) d(x, z)

~

d(x, y) EB d(y, z),

(iv) d(x, y)

=

d( -,x, -,y),

(v) d(x EB s, y EB t)

~

d(x, y) EB d(s, t).

Proof: Properties (i), (ii) and (iv) immediately follow by definition, reealling the basie properties of the natural order on A (Lemmas 1.1.2 and 1.1.4). To prove (iii), note first that -,(x e z) EB (x e y) EB (y e z) = (-,xV-'Y)EB(zVy) ~ -'yEBy= 1. Henee, (xez) ~ (xey)EB(yez). In a similar way we obtain (z e x) ~ (y e x) EB (z e y), whenee (iii) follows from the monotonieity of EB (Lemma 1.1.4(ii)). One similarly proves (v) by observing that -,((x EB s) e (y EB t)) EB (x e y) EB (s e t) = -,(x EB s) EB (x Vy) EB (t V s) ~ -,(x EB s) EB x EB s = 1. 0 As an immediate consequenee we have Proposition 1.2.6 Let I be an ideal of an MV-algebra A. Then the binary relation =1 on A defined by x =1 Y iJ] d(x, y) E I is a eongruenee relation. (Stated otherwise, =1 is an equivalence relation such that x =1 sand y =1 t imply -,x ==1 -,s and x EB y =1 S EB t.)

Moreover, I = {x E A I X =1 O}. Conversely, if is a congruence on A, then {x E A I x == O} is an ideal, and x y iJ] d(x, y) O. Therefore, the correspondence I ~ =1 is a bijection from the set of ideals of A onto the set of congruences on A. 0

=

=

=

16

CHAPTER 1. BASIC NaTIONS

Given x E A, the equivalence class of x with respect to = [ will be denoted by xl1 and the quotient set AI = [ by A11. Since = [ is a congruence, defining on the set AI1 the operations (1.11) ,(xl1) =def ,xl1 and (1.12) xI1$YI1=def (x$y)11, the system (AI1, $",011) becomes an MV-algebra, called the quotient algebra of A by the ideal 1. Moreover, the correspondence x t---t xl1 defines a homomorphism h[ from A onto the quotient algebra A11, which is called the natural homomorphism from A onto A11. Note that Ker(h[) = 1. The next lemma is an easy consequence of Lemma 1.2.3(ii). Lemma 1.2.7 1f A, Band C are MV-algebras, and f: A - Band g:A - C are surjective homomorphisms, then Ker(f) ~ Ker(g) if and only if there is a surjective homomorphism h: B - C such that hof = g, i.e., h(f(x)) = g(x) for alt xE A. This homomorphism h is an isomorphism if and only if Ker(f) = Ker(g). 0 Upon noting that Ker(h)

= Ker(hKer(h») we immediately get

Theorem 1.2.8 Let A and B be MV-algebras. If h: A - B is a surjective homomorphism, then there is an isomorphism f: AI K er(h) - B such that f(xIKer(h)) = h(x) for alt xE A. 0 Proposition 1.2.9 1f A is an MV-chain, then alt proper ideals of A are prime. Proof: Let 1 be a proper ideal of A. Since h[: A - AI1 is a surjective homomorphism, then AI1 is also an MV-chain, and hence, by Lemma 1.2.3(v), 1 must be a prime ideal. 0

Proposition 1.2.10 Let J be an ideal of an MV-algebra A. Then the map 1 t---t hJ(I) determines an inclusion preserving one-one correspondence between the ideals of A containing J and the ideals of the quotient MV-algebra AI J. The inverse map also preserves inclusions, and is obtained by taking the inverse image h;l(K) of each ideal K of AIJ·

1.2. HOMOMORPHISMS AND IDEALS

17

Proo/: Let I be an ideal of A such that J ~ I. Since h J maps A onto AIJ and Ker(h J ) = J ~ I, by Lemma 1.2.3 (ii) and (MV6'), we have hJ(I) E I(AI J) and, moreover, h J1 (h J (I)) ~ I. Since the converse inclusion holds for all surjective mappings, one has that I = h J1 (h J (I)). On the other hand, by Lemma 1.2.3 (i), h J1 (K) E I(A) for each K E I(AI J). To complete the proof it is sufficient to note that J = hJ1 ( {O}) ~ hJ1 (K) and hJ (h J1 (K)) = K. 0 Remark: If A is an MV-chain, then the set I(A) is totally ordered by inclusion. Indeed, if I, J were ideals of A such that I ~ J and J ~ I, then there would be elements a, b in A such that a E 1\ J and b E J \ I, whence a i band b i a, which is impossible. Theorem 1.2.11 The /ollowing properties hold in any MV-algebra A: (i) Every proper ideal

0/ A

(ii) For each prime ideal J ordered by inclusion.

that contains a prime ideal is prime;

0/ A,

the set {I E I(A) I J ~ I} is totally

Prao!" Let J be a prime ideal of A. By Lemma 1.2.3(v), AI J is an MVchain, and by Proposition 1.2.9 and the above remark, all proper ideals of AI J are prime and are totally ordered by inclusion. This, together with Proposition 1.2.10, implies (ii). To prove (i), note that if I is a proper ideal of A such that J ~ I, then again by Proposition 1.2.10, 1= h J1 (h J (I)), whence I is the inverse image of a prime ideal of AI J.

o

Corollary 1.2.12 Every prime ideal J tained in a unique maximal ideal 0/ A.

0/

an MV-algebra A is con-

Prao!" The set 1i. =def {I E I(A) I I =I- A and J ~ I} is totally ordered by inclusion. Therefore, M =def UIE'H I is an ideal of A. Furt her , M is a proper ideal, because 1 ~ M; we conclude that M is the only maximal ideal containing J. 0

The next proposition will play an important role in the proof of Chang's Subdirect Representation Theorem.

18

CHAPTER 1. BASIC NOTIONS

Proposition 1.2.13 Let A be an MV-algebra, J an ideal 0/ A, and a E A \ J. Then there is a prime ideal P 0/ A such that J ~ P and a (j. P. Proo/: A routine application of Zorn's Lemma shows that there is an ideal I of A which is maximal with respect to the property that J ~ land a (j. I. We shall show that I is a prime ideal. Let x and y be elements of A, and suppose that both x 8 y (j. land y 8 x (j. I (absurdum hypothesis). Then the ideal generated by land x8y must contain the element a. By (1.9), a:::; sEIl p(x 8 y) for some sEI and some integer p ~ 1. Similarly, there is an element tEl and an integer q ~ 1 such that a:::; tEIl q(y 8 x). Let u = sEIlt and n = max(p, q). Then u E I, a:::; u EIl n(x 8 y) and a:::; u EIl n(y 8 x). Hence by (1.6) and (1.7), together with Proposition 1.1.6(ii) and Lemma 1.1.8, we have a:::; (uEIln(x8Y)) 1\ (uEIln(Y8x)) = uEIl(n(x8Y) 1\ n(Y8x)) = u, whence a EI, a contradiction. 0

Corollary 1.2.14 Every proper ideal tion 0/ prime ideals. 0

0/ an MV-algebra is

an intersec-

From the above proposition and Corollary 1.2.12 we immediately obtain:

Corollary 1.2.15 Every nontrivial MV-algebra has a maximal ideal.

o

In the next proposition we generalize so me properties of maximal ideals of boolean algebras. Remarkably enough, (the analogues of) these properties do not hold for other extensions of boolean algebras, such as bounded distributive lattices or commutative rings with unit.

Proposition 1.2.16 Let A and B be MV-algebras, and M be a maximal ideal 0/ B. Then we have

(i) For any homomorphism h: A - B, the inverse image h- 1 (M) is a maximal ideal

0/ A;

(ii) Por any subalgebra S

0/ B,

Sn M is a maximal ideal

0/ S.

1.3. SUBDlRECT REPRESENTATION THEOREM

19

Proof" (i) By Lemma 1.2.3, h- 1 (M) is an ideal of A; since h(l) = 1 t/. M, then h- 1 (M) must be a proper ideal. Suppose z t/. h- 1 (M). Since h(z) t/. M, then by Proposition 1.2.2 there is an integer n ~ 1 such that -,nh(z) E M. It follows that -,nz E h- 1 (M), whence, again by Proposition 1.2.2, h- 1 (M) is a maximal ideal of A. The proof of (ii) easily follows from (i), upon letting L: S -- B be the natural embedding, given by L(X) = x for all x E S, and noting that Mn B = L- 1 (M). 0

1.3

Subdirect representation theorem

Throughout this section I shall denote a nonempty set. The direct product of a family {AdiEI of MV-algebras, denoted by TIiEI Ai, is the MV-algebra obtained by endowing the set-theoretical cartesian product ofthe family with the MV-operations defined pointwise. In other words, TIiEI Ai is the set of all functions f: 1-- UiEI Ai such that f(i) E Ai for all i E I, with the operations -, and EI7 defined by

(-,J)(i)

=def

-,f(i) and (f EI7 g)(i)

=def

f(i) EI7 g(i).

The zero element of TI iE1 Ai is the function i E I j E I, the map 7rj: TIiEI Ai -- A j is defined by

t---+

Oi E Ai. For each

f(j)· Each 7rj is a homomorphism onto A j , called the lh projection function. In particular, for each MV-algebra A and nonempty set X, the MV-algebra A X is the direct product of the family {AX}XEX, where A x = A for all x EX.

7rj(f)

=def

Definition 1.3.1 An MV-algebra A is a subdirect product of a family {AihEl of MV-algebras iff there exists a one-one homomorphism h: A -- TIiEI Ai such that for each j EI, the composite map 7rj 0 his a homomorphism onto A j • If Ais a subdirect product of the family {AihEl, then A is isomorphie to the subalgebra h(A) of TIiEI Ai; moreover, the restriction to h(A) of each projection is a surjective mapping. The following result is a particular case of a theorem of Universal Algebra, due to Birkhoff:

20

CHAPTER 1. BASIC NaTIONS

Theorem 1.3.2 An MV-algebra A is a subdirect product of a family {AihEI of MV-algebras if and only if there is a family {JihEI of ideals of A such that (i) Ai ~ AI Ji for each i E I and (ii)

n iEI

Ji = {O}.

Prool Supposing first that A is a subdirect product of a family {Ai hEl of MV-algebras, let h: A ---.. niEI Ai be a one-one homomorphism as given by Definition 1.3.1; for each j E I, let Jj = K er(7rj 0 h). By Theorem 1.2.8, A j ~ AI Jj . If x E niEI J i , then 7rj(h(x)) = 0 for all j E I. This implies that h(x) = 0, and since h is injective, x = O. Therefore niEI Ji = {O}, and conditions (i) and (ii) hold true. Conversely, suppose {JihEI to be a family of ideals of A satisfying conditions (i) and (ii). Let Ei be an isomorphism of AI J i onto Ai, as given by condition (i). Let the function h: A ---.. niEI Ai be defined by stipulating that, for each x E A, (h(x))(i) = Ei(xl J i ). It follows from (ii) that Ker(h) = {O}, whence, by Lemma 1.2.3(iii), h is injective. Since for each i E I the map a E A 1--+ al Ji E AI Ji is surjective, then 7ri 0 h maps A onto Ai' Thus, A is a subdirect product of the family {AihEI, as required. 0

The following result is fundamental: Theorem 1.3.3 (Chang's Subdirect Representation Theorem) Every nontrivial MV-algebra is a subdirect product of MV-chains. Proof: By Theorem 1.3.2 and Lemma 1.2.3(v), an MV-algebra A is a subdirect product of a family of MV-chains if and only if there is a family {~hEI of prime ideals of A such that niEI ~ = {O}. Nowapply Corollary 1.2.14 to the ideal {O}. 0

1.4

MV-equations

As we shall see, an important consequence of Chang's Subdirect Representation Theorem is that in order to prove that an equation holds in

21

1.4. MV-EQUATIONS

all MV-algebras it is sufficient to check that the equation holds in all MV-chains. To give a precise formulation to this result we shall now develop the necessary syntactic machinery.

Definition 1.4.1 By astring (or, word) over a nonempty set S we understand a finite list of elements of S. The latter are often called the symbols of alphabet S. For each natural number t ~ 1, let St =def {O", EB, Xl, ... , Xt, (, An MV-term in the variables Xl, ... , Xt is astring over St arising from a finite number of applications of the following rules:

n.

°

(Tl) The elements and Xi, for i = 1, ... , t, considered as one-element strings, are MV-terms. (T2) If the string

T

is an MV-term, then so is

(T3) If the strings T and

(J'

'T.

are MV-terms, then so is (T EB o)

In other words, astring T over St is an MV-term if and only if there is a formation (or, parsing) sequence for T, Le., a finite list of strings over St, say Tl, ... , Tn , such that T n = T and for each i E {1, ... , n}, Ti satisfies at least one of the following conditions:

(i)

Ti

=

° or

Ti

= Xj, for some 1 :::; j

(ii) there is j < i such that

Ti

:::;

t,

= 'Tj,

(iii) there are j < i and k < i such that

Ti

=

(Tj

EB Tk).

Those strings Ti that belong to every formation sequence for T are said to be the sub terms of T. The following result is known as the unique readability theorem; its proof is precisely the same as for the classical propositional calculus, and is left as a routine exercise.

Theorem 1.4.2 Every term Ti in the variables Xl, ... ,Xn satisfies precisely one of the above conditions (i)-(iii). Moreover, both term Tj of case (ii) and the pair (Tj, Tk) 01 case (iii) are uniquely determined. 0

22

CHAPTER 1. BASIC NaTIONS

We shall henceforth write T(X}, ..• , X n ) to signify that T is an MVterm in the variables Xl, ... , X n . The following strings are examples of MV-terms in the variables Xl, X2, X3:

Notation: Following tradition, for the sake of readability, in writing terms we shall omit the outermost pair of brackets. Further , we shall freely use the symbols 0, 8, V, /\ and 1 to write MV-terms in abbreviated form, in the light of (1.1)-(1.6). Thus, given MV-terms T and (T in the variables Xl, ... , Xt, we shall freely call MV-terms (in the same variables) also such abbreviations as T 0 (T, T 8 (T, T V (T, T /\ (T, and the like.

Definition 1.4.3 Let A be an MV-algebra, T an MV-term in the variables Xl, ... ,Xt, and assurne al, ... , at are elements of A. Substituting an element ai E A for all occurrences of the variable Xi in T, for i = 1, ... ,t, using the unique readability theorem, and interpreting the symbols 0, EB and -, as the corresponding operations in A, we obtain an element of A, denoted TA(al' . .. , at). In more detail, proceeding by induction on the number of operation symbols occurring in T, we define TA(al' .. . , at) as follows: (i)

xf

=

ai,

for each i = 1, ... , t;

(ii) (-,(T)A = -,( (TA); (iii) (0" EB p)A =

(O"A

EB pA).

By the above definition, given any MV-algebra A we can associate each MV-term T in the variables Xl, ... ,Xn with a function TA: An -+ A. Functions arising in this way are called term functions on A. The dependence of TA on n is tacitly understood.

Definition 1.4.4 An MV-equation (for short, an equation) in the variables Xl, ... , Xt is a pair (T, (T) of MV-terms in the variables Xl,· .. ,Xt·

23

1.4. MV-EQUATIONS

Following tradition, we shall write r = a instead of (r, a). An MValgebra A satisfies the MV-equation r = a, in symbols,

A

Fr = a,

Axioms (MV! )-(MV6) are examples of MV-equations in the variables x, y, z. By definition, these equations are satisfied by all MValgebras. In the previous sections many other equations have been shown to hold for all MV-algebras. By contrast, the equation xEBxEBx = x EB x is satisfied by the MV-algebra Ln if and only if n = 2 or n = 3.

°

Remark: By Proposition 1.2.5(i), an MV-equation r = a holds in an MV-algebra A if and only if the equation (r e a) EB (a e r) = holds in A. Therefore we can safely assume that all MV-equations are of the form p = 0, where p is an MV-term. Lemma 1.4.5 Let A, B, Ai (for all i EI) be MV-algebras:

(i) If A

Fr = athen

S

Fr = a

for each subalgebra S of A;

(ii) If h: A - t B is a homomorphism, then for each MV-term r in the variables Xl," ., X s and each s-tuple (al,"" as) of elements of A we have rB(h(al),"" h(a s )) = h(TA(al"'" as)). In particular, when h maps A onto B, from A F T = a it follows that B ~ T

= a;

(iii) If Ai

FT= a

for each i E I, then

I1EI Ai F T = a.

Proof" Conditions (i) and (ii) are immediate. As for condition (iii), let JI, . .. , fs E A = niEI Ai' By hypothesis, for each i E I we can write TA(fI"'" fs)(i) = TAi(JI(i), ... , fs(i)) a Ai (fl (i), ... , fs(i)) = aA(fI"'" fs)(i) ,

=

whence TA(JI, . .. , fs) = aA(JI, . .. ,fs). 0 Theorem 1.4.6 Let A be the subdirect product of a family {AihEl of MV-algebras; let r = a be an MV-equation. Then A F T = a if and only if Ai F T = a for each i EI.

CHAPTER 1. BASIC NaTIONS

24

Proof" Let h: A ~ [liEf Ai be a one-one homomorphism as given by Definition 1.3.1. Suppose that A F r = (J'. Since for each i E I, 7ri 0 h maps A onto Ai, it follows that Ai Fr = (J. Conversely, suppose that Ai F r = (J' for all i EI. By the above lemma, [liEf Ai F r = (J', and since h(A) is a sub algebra of [liEf Ai, h(A) F r = (J'. Since h- 1 maps h(A) onto A, we conclude that A F r

= (J'.

0

Corollary 1.4.7 An MV-equation is satisfied by all MV-algebras if and only if it is satisfied by all MV-chains. Proof" Suppose that r = (J' is satisfied by all MV-chains, and let A be an MV-algebra. If A = {O} then, trivially, rA(O, ... , 0) = 0 = (J'A(O, ... , 0), whence A F r = (J'. If A is nontrivial the desired conclusion follows from Theorems 1.3.3 and 1.4.6. 0

Corollary 1.4.7 will be considerably strengthened in the next chapter, by proving that an equation holds in all MV-algebras if and only if it holds in the algebra [0, 1].

1.5

Boolean alge bras

We have already noted that boolean algebras are particular cases of MV-algebras. In this section we shall characterize boolean algebras among MV-algebras. As shown in Section 1, the natural order makes every MV-algebra A into a lattice with minimum element 0 and maximum 1. We shall denote this lattice by L(A). Recall that the lattice operations of join and meet are definable via the MV-operations by formulas (1.5) and (1.6). A lattice is called distributive Hf the following distributive laws hold:

x 1\ (y V z)

=

(x 1\ y) V (x 1\ z)

and

x V (y 1\ z) = (x V y) 1\ (x V z).

25

1.5. BOOLEAN ALGEBRAS

When A is an MV-chain and a, bE A, then a V b = max(a, b) and a /\ b = min(a, b), whence clearly the distributive laws hold in A. Using (1.5) and (1.6), the above two distributive laws can be equivalently reformulated as MV-equations; since these equations are satisfied byall MV-chains, by Corollary 1.4.7 we obtain Proposition 1.5.1 For any MV-algebra A, L(A) is a distributive lattice with smallest element 0 and greatest element 1. 0 Definition 1.5.2 An element x of a lattice L with 0 and 1 is said to be complemented iff there is an element y E L (the complement of x) such that x V y = 1 and x /\ y = O. When L is distributive each z E L has at most one complement, denoted -z. We further let B(L)

be the set of all complemented elements of the distributive lattice L. Note that 0 and 1 are elements of B(L), because -0 = 1 and -1 = O. As a matter of fact, B(L) is a sublattice of L which is also a boolean algebra. For any MV algebra A we shall write B(A) as an abbreviation of B(L(A)). Elements of B(A) are called the boolean elements of A. Theorem 1.5.3 For every element x in an MV algebra A the following conditions are equivalent:

(i) xE B(A); (ii) x V -,x

= 1;

(iii) x /\ -,x

= 0;

(iv) x E9 x

= x;

(v) x0x=x; (vi) x E9 y

= x V y, for all y E A;

(vii) x 0 y = x /\ y, for all Y E A.

26

CHAPTER 1. BASIC NOTIONS

Proof' The following equivalences are trivial: (ii){:}(iii), (iv){:} (v) , (vi){:}(vii). It is also trivial that (vi) => (iv). Further, the equivalent conditions (ii) and (iii) state that ....,x is the complement of x. Thus, in particular (iii)=> (i). (i)=> (ii): By elementary manipulations, using Lemma 1.1.2 and Proposition 1.1.6 we have ....,x = ....,x EB 0 = ....,x EB (x 1\ -x) = (....,x EB x) 1\ (....,x EB -x) = ....,x EB -x. Thus, -x ~ ....,x and 1 = x V -x ~ x V....,x ~ 1, and we are done. (iii)=>(vi): Using Proposition 1.2.5, together with the Subdirect Representation Theorem 1.3.3 and the inequality xVy ~ xEBy, (which also is an immediate consequence of Theorem 1.3.3) we have d(x EB y,xVy) = (xEBY)0""'(xVy) = (xEBy)0(-,xl\-,y) ~ ((xEBY)0-'x) I\((x EB y) 0 ....,y) = -,x 1\ Y 1\ ....,y 1\ x. Therefore, x 1\ -,x = 0 implies d( x EB y, x V y) = 0, whence x EB y = x V y. (iv)=>(ii): By hypothesis, 1 = ....,x EB x = ....,(x EB x) EB x = -,x V x. 0 Corollary 1.5.4 B(A) is a subalgebra of the MV algebra A. A subalgebra B of A is a boolean algebra iff B ~ B(A). 0 Corollary 1.5.5 An MV-algebra A is a boolean algebra if and only if the operation EB is idempotent, i.e., the equation x EB x = x is satisfied byA. 0 As the reader will recall, for every element z in an MV algebra A, we denote by

(1.13) (z)

=def

{x

E

A I x ~ z EB ... EB z, (n times) far some n > O}

the ideal generated by z.

Corollary 1.5.6 For any MV algebra A and z E A, z is a boolean element of A iff the set {x E A I x ~ z} is an MV ideal iff {x E A I

x::;z} = (z).

Proof' Immediate from Theorem 1.5.3(iv). 0

1.6. MV-GRAINS

1.6

27

MV-chains

In this section we collect several results on totally ordered MV-algebras, to be used in the next chapter. Lemma 1.6.1 The /ollowing properties hold in every MV-chain A:

(i) 1/ xtBy < 1 then x0y = 0; (ii) 1/ x EB y = x EB z and x 0 y = x 0 z then y = z; (iii)

1/ xEBy=xEBz
then y=z;

(iv)

1/ x0y=x0z>0

then y=z;

(v) x EB y = x iff x = 1 or y = 0; (vi) x EB Y = x iff ·x EB'y

.y;

=

(vii) 1/ x EB y = 1 and x EB z < 1 then (x 0 y) EB z = (x EB z) 0 y. Proof: (i) By hypothesis, .x 'i y, whence y < .x. (ii) By hypothesis, max(.x,y) = .xEB(y0x) = .xEB(z0x) = max(-,x,z). Similarly, min( .x, y) = min( .x, z), whence y = z. Condition (iii) is an immediate consequence of (i) and (ii). Condition (iv) follows from (iii) by Lemma 1.1.4(i). Condition (v) follows from (iii). Prom (v) one immediately obtains (vi). Finally, to prove (vii), since by assumption .y :::; x, we get .y EB (x 0 y) EB z = (.y V x) EB z = x EB z < 1 and .y EB (y 0 (x EB z)) = .y V (x EB z) = x EB z, whence (vii) follows from

(iii).

0

Remark: Prom Theorem 1.3.3 it follows that every MV-algebra satisfies conditions (ii) and (vi). Proposition 1.6.2 The /ollowing equations hold in every MV-algebra

A: (1.14) xEByEB(x0y)=xEBy (1.15) (x e y) EB ((x EB .y) 0 y) (1.16) (x 0 y) EB ((x EB y) 0 z)

=

x

= (x 0 z) EB ((x EB z) 0 y).

28

GHAPTER 1. BASIG NaTIONS

Proof: By Theorem 1.3.3 we can safely assume that A is a chain. If x $ y = 1, then (1.14) follows by MV5'). If x $ y < 1, then (1.14) follows from Lemma 1.6.1(i). To prove (1.15), note that if x:S y then xe y = 0 and x = x A Y = (x $ -.y) 0 y; if, on the other hand, y < x then (x e y) $ (x A y) = (x e y) $ y = x V y = x. As aprerequisite for the proof of (1.16) we shall prove the following equation: (1.17) (x 0 y) $ ((x $ y) 0 z) = (x $ y) 0 ((x 0 y) $ z). Indeed, if x$y = 1 both members of (1.17) coincide with (X0Y)$z. If x $ y < 1 then by Lemma 1.6.1(i) both members coincide with (x $ y) 0 z. Thus (1.17) holds for all MV-algebras. Prom MV4) and MV8) we now obtain: (1.18) -.((x 0 y) $ ((x $ y) 0 z))

= (.x 0

.y) $ ((.x $ .y) 0 .z).

To complete the proof of (1.16) we argue by cases as follows: Gase 1: x $ y $ z < 1. Then since A is a chain, by Lemma 1.6.1(i), both members of (1.16) are equal to O. Gase 2: .x $ .y $ .z < 1. Same as Case 1, recalling (1.18). There remains to consider Gase 3: x $ y $ z = 1 and -.x $ -'y $ -.z = 1. Subcase 3.1: x $ y = 1 and x $ z < 1, or x $ y < 1 and x $ z = 1. Then by symmetry, it is sufficient to consider the case x$y = 1 and x$z< 1. Then x0z = 0, and (1.16) becomes (x0Y)$z = (x$z)0y, which follows from Lemma 1.6.1(vii). Subcase3.2: x$y=x$z=1. Then equation (1.16) becomes (1.19) (x 0 y) $ z

= (x 0 z) $ y.

Note that equation (1.19) holds in case x0y = 0 or x0z = O. Indeed, suppose x 0 y = O. Since x $ y = 1, it follows from Lemma 1.1.3 that x = 'y, whence from y = -.x :S z we obtain (x 0 y) $ z = z = y V z =

1.7. BIBLIOGRAPHICAL REMARKS

29

(-,y 0 z) EEl y = (x 0 z) EEl y. By symmetry, (1.19) holds under the hypothesis x 0 z = O. We next observe that if one of the members of (1.19) is equal to 1 then so is the other. Assume, for instance, (x 0 y) EEl z = 1. Since -.x EEl-.y EEl-.z = 1 is equivalent to x 0 Y 0 z = 0, it follows from Lemma 1.1.3 that z = -.(x 0 y) = -.x EEl -.y. Hence, by Proposition 1.1.6, (x 0 z) EEl y = (x 0 (-,x EEl -.y)) EEl y = (x /\ -,y) EEl y = (x EEl y) /\ (-,y EEl y) = 1. Thus, to complete the analysis of Subcase 3.2 we may restriet to the case when (x 0 y) EEl z < 1, (x 0 z) EEl y < 1, x 0 y > 0, x 0 z > O. Under these hypotheses, by Lemma 1.6.1(vii) we obtain:

x0 (zEEl (x0Y)) = (x0z) EEl(x0y) > 0, x 0 (y EEl (x 0 z)) = (x 0 y) EEl (x 0 z), thus establishing (1.19) by Lemma 1.6.1(iv). Subcase 3.3: x EEl y < 1 and x EEl z < 1. Then by Lemma 1.6.1(i), x0y = 0 and x0z = 0, i.e., -,xffi-,y = 1 and -,x$-.z = 1. Recalling (1.18) and arguing as in Subcase 3.2 (with -,x, -'y, -,z instead of x, y, z) we conclude that (1.16) also holds in this case. 0

1.7

Bibliographical remarks

MV-algebras were originally introduced by Chang in [36] to prove the completeness theorem for the Lukasiewicz calculus. In the same paper Chang proved the basic facts about the natural order, congruences and boolean elements. Prime ideals were introduced by hirn in [38] to prove Theorem 1.3.3. For historical information on MV-algebras see [40]. A version of Proposition 1.2.2 is in [86]. Equation (1.16) was proved in [51]. It is easy to see that the present definition of MV-algebras (which is essentially due to Mangani [144]) is equivalent to Chang's original definition. To this purpose one simply not es that for every MV-algebra (A, $, -', 0) the operations V and /\ defined by (1.5) and (1.6) are the join and the meet with respect to the natural order of A; thus in particular, both operations are commutative and associative.

CHAPTER 1. BASIC NaTIONS

30

(Equivalents of) MV-algebras are known in the literat ure under sever al names. As an example, following [249],[121], let us say that a bounded commutative ECK-algebra is an algebra (A, *, 0,1) with a binary (bounded subtraction) operation * and two constants 0 and 1 satisfying the following equations: Y1) (x * y) * z = (x * z) * y

Y2) x*(x*y)=y*(y*x) Y3) x*x = 0 Y4) x*o= x Y5) x

*1 =

O.

Bounded commutative BCK-algebras have been considered by several authors. See, e.g., [233], [213], [214], [116], [120]. A tedious but straightforward verification yields the following result, first proved in [86] (also see [164]):

Theorem 1.7.1 If (A, EB,', 0) is an MV-algebra then (A, e, 0,1) is a bounded commutative ECK-algebra. Moreover, we have the identities .x = 1 ex and x EB y = 1 e ((1 e x) e y)). Conversely, for any bounded commutative ECK-algebra (A, *,0,1), upon dejining .x =dej 1 * x and x EB y =dej 1 * ((1 * x) * y), then (A, EB,', 0) is an MV-algebra, and x e y = x * y. There exist several other equivalent counterparts of MV-algebras, including Bosbach's bricks [31], Buff's S-algebras [33], Komori's CNalgebras [130], Lacava's L-algebras [131], Rodriguez's Wajsberg algebras [212], [86]. We will return to the latter in subsequent chapters. As we shall also see, up to categorical equivalence, MV-algebras are the same as abelian lattice-ordered groups with a distinguished strong unit. Thus MV-algebras provide an equational formulation of the theory of magnitudes with an archimedean unit. Among all associative commutative continuous operations on the unit interval [0,1], the EB operation and its dual 0 have a special status. For details we refer, e.g., to the papers by Menu and Pavelka [153], [201], and to the relevant chapters ofthe book by Butnariu and Klement [34].

Chapter 2 Chang completeness theorem In this chapter we shall prove Chang's completeness theorem stating that if an equation holds in the unit real interval [0, 1], then the equation holds in every MV-algebra. Thus, intuitively, the two element structure {O, 1} stands to boolean algebras as the interval [0,1] stands to MValgebras. Our proof is elementary, and makes use of tools (such as "good sequences") that shall also find applications in a subsequent chapter to show the equivalence between MV-algebras and lattice-ordered abelian groups with streng unit.

2.1

The functor

r

A partially ordered abelian group is an abelian group (G, +, -,0) endowed with a partial order relation::; that is compatible with addition; in other words, ::; has the following translation invariance property, for all x, y, t E G: (2.1)

If x::;y then t+x::;t+y.

°: ;

The positive cone G+ of G is the set of all x E G such that x. When the order relation is total, (Le., when G = G+ U -G+), G is said to be a totally ordered abelian group, or o-group for short. When the order of G defines a lattice structure, G is called a lattice-ordered abelian group, or f-group, for short. In any f-group we have

(2.2) t+(xVy)=(t+x)V(t+y) 31 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

32

CHAPTER 2. CHANG COMPLETENESS THEOREM

and

(2.3)

t+(xAy)=(t+x)A(t+y).

In every o-group xVy = max{x,y} and xAy = min{x,y}; o-groups are particular cases of f-groups. For each element x of an f-group G, the positive part x+, the negative part x-, and the absolute value lxi of x are defined as folIows:

(2.4)

x+ =d,eJ 0 V x;

(2.5)

x- =d,eJ 0 V -x;

(2.6)

lxi =d,eJ x+ + x- = x

V

-x.

A strong (order) unit u of G is an archimedean element of G, Le., an element 0 ~ u E G such that for each x E G there is an integer n ~ 0 with lxi ~ nu. Definition 2.1.1 Let G be an f-group. For any element u E G, u > 0, (not necessarily u being a strong unit of G) we let

[0, u] =d,eJ {x

E G

I 0 ~ x ~ u},

and for each x, y E [0, u],

x E9 y =d,eJ u A (x + y),

and -,x =d,eJ u - x.

The structure ([0, u], E9, -,,0) is denoted r(G, u). Proposition 2.1.2 r(G, u) is an MV-algebra. Proof: We shalllimit ourselves to verifying that r(G, u) satisfies MV6). For all x, y E [0, u] we have

-,( -,x E9 y) E9 y

= y E9 -,(y E9 -,x) = u A (y + (u - (u A (y + u - x))))

2.1. THE FUNCTOR = u A (y

r

33

+ u + (-u V (-y -

= u A «y + u - u) V (y

u

+u -

+ x))) Y- u

+ x))

=uA(yVx)

= y V x = x Vy. This shows that x and y are interchangeable. 0

Lemma 2.1.3 Let G be an f-group with strong unit u. r(G,u).

Let A -

(i) Forall a,bEA,a+b=(aEDb)+(a0b); (ii) For all

Xl, ... , X n E

A,

Xl

ED ... ED X n

= U A (Xl + ... + X n );

01 the MV-algebra A coincides with the order inherited from G by restrietion.

(iii) The natural order

01 [0, u]

Proof" (i) We easily obtain a + b - (a 0 b) = a + b - (0 V (a + b - u» = (a+b) Au = aEDb. An easy induction on n proves (H). (iii) is proved by direct verification, using 1.1.5, together with the above proof of 2.1.2.

o

Notation: Following common usage, we let R, Q, Z denote the additive groups of reals, rationals, integers, with the natural order. In the particular case when G = R, r(R, 1) coincides with the MV-algebra [0,1]. We also have Q n [0,1] = r(Q, 1). Further, for each integer n 2: 2, 1 1 z Ln = r(Z n _ 1,1), where Z n _ 1 = {n _ 1 I Z E Z}.

In particular, the boolean algebra {O, 1}

= L 2 is the same as r(Z, 1).

When dealing with elements X E r(G,u), the notation nx may ambiguously represent both X + ... + X (n times), and X ED ... ED X (n times). To avoid any danger of confusion, we will adopt the following notation: (2.7)

n.x =def "-----v---" X ED··· ED X = u A (x n times

+ ... + x), = n ti~es

u A nx.

34

CHAPTER 2. CHANG COMPLETENESS THEOREM

Let G be an f-group and 0< u E G. Let S = {x E GI for so me O:S: n E Z, lxi :s: nu}. Then S is a subgroup and a sublattice of G containing u, and r(G, u) = r(S, u). Therefore, when considering the MV-algebras r(G, u) we can safely assurne that u is a strong unit of G. Definition 2.1.4 Let G and H be f-groups. A function h: G ----t H is said to be an f-group homomorphism iff h is both a group-homomorphism and a lattice-homomorphism; in other words, for each x, y E G, h(x-y) = h(x)-h(y), h(xVy) = h(x)Vh(y) and h(x/\y) = h(x)/\h(y). Suppose that 0 < u E G and 0 < v E H, and let h: G ----t H be an fgroup homomorphism such that h(u) = v. Then h is said to be a unital f- homomorphism. Letting r(h) =def hho,u] denote the restrietion of h to the unit interval [0, u], then r(h) is a homomorphism from r(G, u) into r(H, v). Proposition 2.1.5 Let Adenote the category whose objects are pairs (G, u) with G an f-group and u a distinguished strang unit of G, and whose morphisms are uni tal f-homomorphisms. Then r is a functor from A into the category MV of MV-algebras. 0 The above result shall be strengthened in a subsequent chapter, where we shall prove that r is a natural equivalence (Le., a full, faithful, dense functor) between A and MV.

2.2

Good sequences

A sequence a = (al, a2, .. .) of elements of an arbitrary MV-algebra A is said to be good iff for each i = 1,2 ... ,

and there is an integer n such that ar = 0 for all r > n. Instead of a = (al, ... , an, 0, 0, ... ) we shall often write, more concisely,

35

2.2. GOOD SEQUENCES Thus, if quences

om

denotes an m-tuple of zeros, we have identical good se-

Note that by pre-pending to the good sequence (ab" . ,an) an m-tuple 1m of consecutive ones, the resulting good sequence (1 m , al, ... ,an) is different from (al,"" an). For each a E A, the good sequence (a,O .. . ,0, ... ) will be denoted by (a). The good sequences of a boolean algebra Aare the nonincreasing sequences of elements of A having a finite number of nonzero terms. For totally ordered MV-algebras we have the following characterization of good sequences: Proposition 2.2.1 If A is an MV-chain then each good sequence of A has the form

(2.9)

(F, a)

for some integer p ~ 0 and a E A.

Proof: Immediate from Lemma 1.6.1(v). 0

Lemma 2.2.2 Suppose that A ~ TIi Ai is the subdirect product of a family {AihEl of MV-algebras. A sequence a = (al,"" an, .. ') of elements of A is a good sequence if and only if for each i E 1 the sequence

is a good sequence in Ai, and there is an integer no > 0 such that whenever n > no then for all i E I, '7ri(an ) = O. Prao!: It is sufficient to note that an EB an+l an+!) = '7ri(an ) for each i E I. 0

=

an means that '7ri(a n EB

In the light of Theorem 1.3.3, the above Lemma 2.2.2 and Proposition 2.2.1 yield a very useful tool for dealing with good sequences. As an example, consider the proof of the following result:

Lemma 2.2.3 Let A be an MV-algebra. If a = (al,"" an, . .. ) and b = (bI,' .. , bn , . .. ) are good sequences of A, then so is the sequence c = (Cl, ... , Cn , . .. ) given by Cn = an V bn for each n.

36

CHAPTER 2. CHANG COMPLETENESS THEOREM

Proof: Since a and bare both good sequences, there is an integer no such that en = 0 for all n > no. By Theorem 1.3.3, A is a subdireet product of a family {CihEl of MV-chains. For each i E I the sequences ~ = (7ri(al)"'" 7ri(an), ... ) and b i = (7ri(bd, ... , 7ri(bn), ... ) are good sequences of Ci' Hence, by Proposition 2.2.1, ~ = (l P , ai) and b i = (l Q ,ßi), where ai and ßi are in Ci' Therefore, 7ri(en) = 1 if n :5 max{p, q} and 7ri(en) = 0 if n > max{p, q} + 1. For n = max{p, q} + 1, we have 7ri(en) = ai if p > q, 7ri(Cn) = ßi if P < q and 7ri(en) = max{ai,ßi} when p = q. Consequently, letting Ci = (7ri(CI), ... , 7ri(en), ... ) it follows that Ci is a good sequence for each i E I, whence we conclude that C is a good sequence of A. 0

Example: To have a better intuition of the meaning of good sequences, for every real number a ~ 0 let laJ denote the greatest integer :5 a, and let (a) =def a - laJ be the fractional part of a. There will be no danger of confusion between this notation and the notation for principal ideals. Then a can be written as a = 1 + ... + 1 + (a)

+ 0 + 0 + ...

with laJ many consecutive l's. Considered as elements of the MValgebra [0,1], the above summands al, a2,' .. of a trivially satisfy the identity ai EB ai+l = ai for every integer i ~ 1. For 0 :5 ß E R, let similarly

ß = ßI + ... + ßm-l whereßI = ... = and 0 = ßm+1 = where "11 = ... = and 0 = "In+m+l each i = 1,2, ... , (2.10) "Ii

+ (ß) + 0 + ... ,

ßm-l = 1 = al = ... = an-I, 0 = an+l = a n+2 = ... , ßm+2 = .... Let "I = a + ß· Then "I = "11 + "12 + ... , "In+m-2 = 1, "In+m-l = (a) EB (ß), "In+m = (a) 0 (ß), = "In+m+2 = .,. . In a more compact notation, for the summand "Ii is given by

= aiEB(ai-10ßI)EB(ai-20ß2)EB .. .EB(a20ßi-2)EB(a10ßi-l)EBßi'

In the light of (2.10) and (2.8), we can now give the following Definition 2.2.4 For any two good sequences a = (al"", an) and b = (bI, ... , bm ) their sum C = a + b is defined by C = (Cl, C2,.· .), where for all i = 1,2, ...

(2.11)

Ci =def ai EB (ai-l 0 bd EB ... EB (al 0 bi-d EB bio

2.3. THE PART1ALLY ORDERED MON01D M A

37

Since ap = bq = 0 whenever p > n and q > m, then Cj identically vanishes for each j > m + n. The notation c = (Cl,"" Cn+m) = (al,' .. ,an) + (bI, ... , bm ) is self-explanatory. The following immediate consequence of (2.11) will be frequently used to compute the sum of two good sequences in an MV-chain:

2.3

The partially ordered monoid MA

Since by equation (1.14), (a EB b, a 0 b) is a good sequence, applying Theorem 1.3.3 and Lemma 2.2.2 together with (2.12), we immediately get that the sum of two good sequences is a good sequence. We denote by M A the set of good sequences of A equipped with addition.

Proposition 2.3.1 Let A be an MV-algebra. Then M A is an abelian monoid with the following additional properties: (i) (cancellation) For any good sequences a, b, c, if a + b = a then b = c;

+c

(ii) (zero-Iaw) 1f a + b = (0) then a = b = (0). Proof: By (2.11), a+(O) = a, addition is commutative, and the zero-law holds. To prove associativity, by Theorem 1.3.3 we can safely assume A to be totally ordered. By Proposition 2.2.1 and equation (1.16) in Proposition 1.6.2, letting a = (l P , a), b = (F, b), and c = (F, c), we have the identities

(b + a)

+c

= (lP+q+r, aEBbEBc, (a0b)EB«aEBb)0c), a0b0c) = (lP+q+r,

a EB bEB c, (a 0 c) EB «a EB c) 0 b), a 0 b 0 c)

=b+(a+c). Similarly, to prove cancellation, avoiding trivialities, assume that a, band c are different from 1. If q = r, then by Lemma 1.6.1(ii), b = c,

38

CHAPTER 2. CHANG COMPLETENESS THEOREM

and we are done. If q < r-1 then from the identity (1 p +q , aEBb, a0b) = (1 P+r ,a EB c,a 0 c) we get a 0 b = 1, Le., a = b = 1, which is a contradiction. If q = r - 1 then a 0 b = a and a EB b = 1, which is impossible because these two equalities imply that b = 1. The cases corresponding to r < q are similarly shown to lead to contradiction. 0 Proposition 2.3.2 Let a = (al, ... , an) and b = (bI, ... , bm) be good sequences. Recalling (2.8) assume, without loss of generality, m = n. Then the following are equivalent:

(i) There is a good sequence c such that b (ii) bi

~

for all

ai

+c = a

,.

i = 1, ... ,n.

Praof: (i):::} (ii) is immediate from (2.11). (ii):::} (i). Observe that by the remark following Lemma 1.6.1, (-.bn , ... , -.b l ) is a good sequence. Let us denote by c = a - b the good sequence obtained by dropping the first n terms in (al,"" an) + (-.b n, ... , -.b l ). We shall prove that c + b = a. Using Theorem 1.3.3 we can safely assume A to be totally ordered, so that by (2.9), a = (1 P , a) and b = (1 q , b). To avoid trivialities assume both a and b to be different from 0 and from 1. Then q ~ p. Upon rewriting b = (1 Q, b, QP-Q) , from n = p + 1 we get (-.bn , ... , -.bd = (F-Q, -.b, oQ), and hence c is obtained by dropping the first p + 1 terms from (1 2p - Q, a EB -.b, a e b). Gase 1: b ~ a. Then a EB -.b

= 1, c = (lp-q,a e b) and c + b = (1 P , (a e b) EB b) = (1 P , b V a, 0) = (1 P , a) = a.

b, (a e b) 0

Gase 2: b> a. Then p > q, aeb = 0, c = (1 P - q - l , aEB-.b) and c+b (1 P - I , a EB -.b EB b, (a EB -.b) 0 b) = (lP, a!\ b) = (F, a) = a. 0

Definition 2.3.3 Given any two good sequences a and b of A we write:

(2.13) b

~

a

iff band a satisfy the equivalent conditions of 2.3.2.

2.3. THE PARTIALLY ORDERED MONOID M A

39

Proposition 2.3.4 Let a and b be good sequences. (i) If b $ athen there is a unique good sequence c such that b+c = a. This c, denoted a - b, is given by

(ii) In particular, for each a E A we have (2.15) (-,a) = (1) - (a). (iii) The order is translation invariant, in the sense that b $ a implies b + d $ a + d for every good sequence d. Proof" By an easy adaptation of the proof of Proposition 2.3.2, together with Proposition 2.3.1 (i). 0

Proposition 2.3.5 Let a = (al, . .. , an, ... ) and b = (bI, ... ,bn , ... ) be good sequences of an MV-algebra A. (i) The sequence

is good, and is in fact the supremum of a and b with respect to the order defined by (2.13). (ii) Analogously, the good sequence

is the infimum of a and b.

(iii) For all a, b, c E A we have

(2.16)

((a)

+ (b)) /\ (1) = (a $

b).

Proof" By Lemma 2.2.3, together with Proposition 2.3.2(ii) and (2.11).

o

40

2.4

CHAPTER 2. CHANG COMPLETENESS THEOREM

Chang's f-group GA

From the abelian monoid M A , enriched with the lattice-order of Proposition 2.3.5, one can routinely obtain an f-group GA such that M A is isomorphie, both as a monoid and as a lattice, to the positive cone GA +. To this purpose, mimicking the construction of Z from N, let us agree to say that a pair of good sequences (a, b) is equivalent to another pair (a', b') iff a + b' = a' + b. Transitivity of this relation follows from cancellation, Proposition 2.3.1(i). Notation: The equivalence class of the pair (a, b) shall be denoted by

[a, b]. There will be no danger of confusion with the notation for unit intervals in f-groups. Let GA = (GA, 0, +, -) be the set of equivalence classes of pairs of good sequences, where the zero element 0 is the equivalence class [(0), (0)], addition is defined by [a, b]

+ [c, d] =def [a + c, b + d],

and

subtraction is defined by -ra, b] =def [b, a].

Then by direct inspection one easily sees that GA is an abelian group. GA is called the enveloping group of A. We shall now equip GA with a lattice-order. Let (a, b) be a pair of good sequences of the MV-algebra A. By Proposition 2.3.2(i), (a, b) has an equivalent pair of the form (e, (0)) if and only if a 2: b. Let M~ be the submonoid of GA given by the equivalence classes of pairs (e, (0)), for all good sequences e. Since the map e 1---+ (e, (0)) induces an isomorphism of the monoid M A onto M A, we shall freely identify the two monoids M A and M A. Definition 2.4.1 Let A be an MV-algebra, and a, b, c, d E M A . We say that the equivalence class [c, d] dominates the equivalence class [a, b], in symbols, [a, b] :::5 [c, d],

2.4. CHANG'S i-GROUP GA

41

Hf [C, d]-[a, b] = [e, (0)] for some good sequence e E M A . Equivalently, [a, b] ~ [c, d] Hf a + d ~ c + b, where ~ is the partial order of M A given by Definition 2.3.3. Proposition 2.4.2 Let A be an MV-algebra. (i) The relation ~ is a translation invariant partial order, making GA into an i-group. Specijically, for any two pairs of good sequences (a, b) and (c, d) the supremum of their equivalence classes in GA is the equivalence class of ((a + d) V (c + b), b + d), where V is the supremum in M A given by Proposition 2.3.5. In symbols,

(2.17) [a, b] V[c, d] = [(a + d) V (c + b), b + d]. (ii) Similarly, the injimum [a, b] A[c, d] is given by (2.18) [a, b] "[c, d]

= [(a + d) /\ (c + b), b + d].

(iii) The map a E M A t-+ [a, (0)] is an isomorphism between the monoid M A , equipped with the lattice-order of Proposition 2.3.5, and the positive cone GA + =def {[c, d] E GA I c 2: d}, with the lattice-order inherited by restriction of~. Proof: (i) The proof that

~

is a translation invariant partial order on

GA is routine. In order to prove (2.17), first of all, from the inequality a+d ~ (a+d)V(c+b) weobtain [a,b] ~ [(a+d)V(c+b),b+d], and, symmetrically, [c,d] ~ [(a+d)V(c+b),b+d]. Thus, [(a+d)V (c + b), b + d] is an upper bound of [a, b] and [c, d]. To show that this is indeed the least upper bound, for any upper bound [p, q] we must find an element z E M A such that

(2.19) p + d + b

= z + q + ((a + d) V (b + c)).

By hypothesis, there are x, y E M A such that p + b = x + q + a and p + d = y + q + c. Let z E M A be such that x + y = z + (x V y); the existence of z is ensured by the inequality x + y 2: x V y, using Propositions 2.3.2 and 2.2.3. One now establishes (2.19) using the cancellation property of M A , as folIows:

2p+ b+d

42

CHAPTER 2. CHANG COMPLETENESS THEOREM

= 2q + a + c + x + Y = 2q + a

+ c + z + (x V y) = z + q + ((x + q + a + c) V (y + q + a + c)) = z + q + ((p + b + c) V (p + d + a)) = p + z + q + ((b + c) V (d + a)). One similarly proves (ii). Finally, (iii) is an immediate consequence of the definitions of the partial orders $ and ::S. 0

Definition 2.4.3 The f-group GA with the above lattice-order is called the Chang f-group of the MV-algebra A. Proposition 2.4.4 The element the f-group GA.

UA

=del [(1), (0)] is a strong unit

0/

Proof: As a matter of fact, any element of GA + can be represented by [a, (0)], for some good sequence a = (al, a2,' .. ) in A. Let the integer m ~ 1 be so chosen that an = 0 for all n ~ m. By Definition 2.4.1, mUA = [1 m , (0)] dominates [a, (0)], whence the desired conclusion immediately follows. 0 A crucial property of the f-group GA is given by the following result:

Theorem 2.4.5 The correspondence

a I-t 'PA(a) = [(a), (0)] defines an isomorphism from the MV-algebra A onto the MV-algebra

r(GA,UA)' Proof: By definition, [(0), (0)] ::S [a, b] ::S UA iff there is c E A such that (a, b) is equivalent to ((c), (0)). Thus, 'PA maps A onto the unit interval [[(0), (0)], UA] of GA' It is easy to see that this map is one-one. By (2.16), 'PA(aEBb) = ('PA(a)+'PA(b)) /\UA, and by (2.15), 'PA(--,a) = UA -'PA(a). Therefore, 'PA is a homomorphism from A to r(GA,UA).

o

Remark: An MV-algebra A is a chain if and only if GA is totally ordered. Indeed, if A is totally ordered, then it follows from Proposition 2.3.2(i) that M A is totally ordered, and this implies that GA is a totally ordered group. The converse is an immediate consequence of the Theorem 2.4.5 above.

2.5. CHANG COMPLETENESS THEOREM

2.5

43

Chang completeness theorem

An f-group term in the variables Xl, •.. ,Xt is astring of symbols over the alphabet {Xl,"" X n , 0, -, +, V, A, (,)} which is obtained by the same inductive procedure used in Chapter 1.4 to define MV-terms. Let T be an f-group term in the variables Xl,' .. , Xt and G be an f-group. Substituting an element ai E G for all occurrences of the variable Xi in T, for i = 1, ... , t, and interpreting the symbols 0, -, +, V and A as the corresponding operations in G, we obtain an element of G, denoted TG(aI, ... , at). To each MV-term T in the n variables Xl, ... , X n we associate an f-group term f in the n + 1 variables (Xl,"" X n , y), according to the following stipulations: Xi =def Xi,

for each i

=

1, ... , n,

Ö=def 0,

=:;a =def (y - a), (p EB(1)

=def

(y

A

(,0 + a)).

Since unique readability also holds for f-group terms, the mapping T I--t f is weIl defined; indeed, a moment's reflection shows that this map is computable by a Turing machine. We then have a purely syntactic counterpart of the mappings (G,u) I--t r(G,u) and A I--t GA, in a sense that is made precise by the following two propositions:

°: ;

°

Proposition 2.5.1 If G is a totally ordered abelian group, < u E G, gl, ... gn ::; u and A = r(G,u), then for every MV-term T(XI, .. . , x n ) we have TA(gl'" ., gn) = fG(gl'" ., gn, u).

Proof: By induction on the number of operation symbols in T. The basis is trivial. For the induction step, by definition of r we have:

(-,(1 )A(gl' ... ,gn)

= -'((1A(gl"" ,gn))

44

CHAPTER 2. CHANG COMPLETENESS THEOREM

The EB-case is similar. 0 In the light of Proposition 2.4.2(iii), we shall now identify M A with the positive cone of GA : Proposition 2.5.2 If A is an MV-chain, al,"" an are elements of A, and r(xI, ... , x n) is an MV-term, then the one-term sequence (rA(al"'" an)) E MA ~ G~ coincides with fGA((ad,···, (an), (1)).

Proof" By induction on the number of operation symbols in r. The basis is trivial. For the induction step, if a = EB'l/J then using (2.16), together with the definition of the mapping a""'" iJ, and omitting unnecessary superscripts, we can write:

= min((l), ((aI, ... , an)) + ('l/J(al"'" an))) = min((l), ~((al)'" ., (an), (1)) + {b((ad,· .. , (an), (1)))

= (y /\ (~+ {b))((al)"'" (an), (1)) = EB'l/J((ad,···, (an), (1)). In the -,-case, one similarly uses (2.15). 0 Theorem 2.5.3 (Completeness Theorem) An equation holds in [0,1] if and only if it holds in every MV-algebra.

45

2.5. CHANG COMPLETENESS THEOREM

Prool Suppose an equation fails in an MV-algebra A. By the remark following Definition 1.4.4 we mayassume that the equation has the form r(xI' ... ' x n ) = o. By Corollary 1.4.7, A may be assumed to be totally ordered. There are elements al, ... ,an E A such that rA(al' . .. , an) > o. Letting GA denote the Chang f-group of A, and again writing M A = G!, by Proposition 2.5.2 we have 0 < fGA((ad,···, (an), (1)) ~ (1). Let S = Z(l) + Z(ad + ... + Z(a n ) be the subgroup of GA generated by the elements (1), (al), ... ,(an), with the induced total order. Since the order in GA is translation invariant, it follows that GA is torsion-free. Since S is a finitely generated subgroup of GA, by the fundamental theorem on torsion-free abelian groups, we can identify S with the free abelian group zr, for some integer r ;::: 1. Its elements (1), (al), ... , (an) are respectively identified with vectors h o, h l , ... ,hn E zr; the set of nonnegative elements of S then becomes a submonoid P of zr such that

(2.20)

P n -P = {O}

and

PU -P = Zr.

For any two vectors h, k E zr let us write h

~p

Let us display the subterms

k

iff

k - h E P.

0"0,0"1, ... ,O"t

of f as follows:

We can safely assurne that the list contains the zero term. The map y ~ h o, Xl ~ h l ,· .. , X n ~ h n uniquely extends to an interpretation O"j ~ h j (j = 0, ... ,t) of subterms of f into elements of the totally ordered group T = (zr, ~p). In particular, by hypothesis we have (2.21) (2.22)

o ~p h l , ... , h n ~p h o, 0 ~p h t o=f. h t = fT(h l , ... , h n , h o).

~p

h o,

Let w be apermutation of {O, ... , t} such that (2.23)

hw(o)

~p

hw(1)

Our aim is to replace

~p ... ~p

hw(t).

by another total order ~P' over zr in such a way that the above inequalities still hold with respect to ~P" ~p

46

CHAPTER 2. CHANG COMPLETENESS THEOREM

and (zr, $;p') is isomorphie to a subgroup of the additive group R with the natural order. For each j = 1, ... , t, let the vector d j E P be defined by

= hw(j)

dj

-

hw(j-I)

= (djll dj2 , ... , djr).

Embedding zr into R r , we define the positive and the negative span of the d j 's as follows: t

(2.24)

P'" =

{E Ajdj I 0 $; Aj ER}, j=I

N'" = -P"'.

Note that P'" is a closed and convex subset of Rr, and whenever h E P* and 0 $; a E R, then ah E P*. Claim 1: Whenever Al, ... , At are real numbers 2: 0 and E~=I Aidi = 0, it follows that Ai = 0 for each i such that d i =I- O. Otherwise (absurdum hypothesis ) let I = {i E {I, ... , t} I d i =IO}, and assurne i'" E I, Ai- > 0 and EiEI Aidi = 0 for suitable Ai 2: O. Stated otherwise, the homogeneous system of linear equations EiEI Aidik = 0, k = 1, ... ,r has nontrivial solutions Ai 2: O. Now, the solutions of this system are obtained by fixing arbitrarily some values, say Aal"'" Aal" with p strictly smaller than the number of elements of I, and then, for each ß E 1\ { aI, ... , a p }, computing Aß by means of the formulas Aß = Ef=I CßaiAai' Here, the coefficients Cßai arise as the result of performing rational operations on the integers dij . Thus, by continuity, the existence of positive real solutions of the system implies the existence of rational positive solutions. Choosing all Ai rationals and multiplying by their least common denominator, we finally obtain integers {ni 2: 0 I i E I} such that EiEI nidi = 0, and ni- 2: 1. By definition of P, we have 0 =I- d i - $;p EiEI nidi = 0, whence d i - = 0, a contradiction.

Having thus settled our first claim, it easily follows that (2.25)

P*

n N* = {O}

From equations (2.20) and (2.25) we obtain P* n P - P* n whence in partieular, for arbitrary i, j = 0, ... , t,

zr,

2.5. CHANG COMPLETENESS THEOREM

47

To conclude the proof we need the foIlowing weIl known result of linear algebra, whose proof is included for the sake of completeness. Here, as usual, h . k denotes the scalar product of vectors h, k E RT. Glaim 2: Let el, ... ,em be vectors in RT such that for any sequence of real numbers 0 ~ Ab ... , Am, if E~IAiei = 0, then Al = ... = Am = O. Then there is a vector V E RT such that ei . V > 0 for each i = 1, ... , m. The proof is by induction on m. The basis is trivial. For the induction step, let el, ... ,em+1 in RT have the property that for any real numbers 0 ~ Ab ... , Am +l, ifE~tlAiei = 0, then Al = ... = Am +l = O. A fortiori, for any 0 ~ 111, ... , 11m with E~l l1iei = 0, we must have 111 = ... = 11m = O. Then by induction hypothesis there is a vector u E RT such that ei· u > 0 for each i = 1, ... , m. We now argue by cases: Gase 1: e m+1· u > O. Then the desired conclusion follows upon letting v=u.

e m+l· u = O. Then one gets the desired result by letting V = au + em+1, where a = 1 + max{ei~:~~t li = 1, ... , m}. Gase 3: e m+l· u < O. Then for each i = 1, ... , mIet the vector gi be defined by gi = ei em+l· e m +l· U By our assumptions ab out the vectors el, ... , e m+1, whenever Vi 2: 0 and E~1 Vigi = 0 then necessarily, VI = ... = Vm = O. By induction hypothesis, there is a vector t E RT such that gi·t > 0 for i = 1, ... , m. Let w be defined by em+l . t w= t u. e m+l· U Then, by direct inspection, w satisfies the inequalities em+l . w = 0 and ei . w > 0 for each i = 1, ... ,m. Proceeding now as in Case 2 we get the desired conclusion, and the claim is settled. Gase 2:

It follows from Claims 1 and 2 that there is a vector

such that g. d j > 0 for all nonzero vectors d j , j = 1, ... , t. By continuity, g can be assumed to be in general position, in the sense

CHAPTER 2. CHANG COMPLETENESS THEOREM

48

that 1'1,"" I'r are linearly independent over Q. Let

Then from (2.24) it follows that (2.27) P* ~

rr;

and N* ~

-rr;.

Let us now focus attention on the totally ordered abelian group T'

= (zr, '5:.P' ). Although T' need not coincide with T = (zr, '5:.p), still by (2.25)-(2.27), for all i,j = 0, ... , t such that h i =1= h j we have

For any vectors ko, ... , kn E zr, the map y t-+ ko, Xl t-+ k I , ... , X n t-+ kn uniquely extends to an interpretation O'j t-+ k j , j = 0, ... ,t of all subterms O'j of f into elements k j of T'. In the particular case when ko = h o, ... , kn = h n , arguing by induction on the number of operation symbols occurring in O'j, from (2.28) we obtain k j = h j for all j = 0, ... , t; moreover, all inequalities in (2.21) are still valid with respect to the new total order relation '5:.p' over zr. In symbols, 0 '5:.p' h I , ... ,hn '5:.p' h o, and 0 '5:.p' h t '5:.p' h o, 0 =1= h t = fT' (h I , ... ,hn , h o)· As an effect of the independence of the I"S over Q, T' is isomorphie, as a totally ordered group, to the subgroup U = ZI'I + ... + Zl'r of R generated by 1'1,"" I'r, with the natural order. An isomorphism is given by the map

Since the inequalities in (2.21) are preserved under isomorphism, letting ~o = O(ho), ~I = O(hd,···, ~n = O(hn ), ~t = O(h t ) we can write 0 '5:. ~I, ... ,~n '5:. ~o and 0 < ~t '5:. ~o. Assuming without loss of generality, ~o = 1, we have ~t = fU(~I"",~n,1) > O. By Proposition 2.5.1, in the MV-algebra B = r(U, 1) we have TB(~I"'" ~n) =1= 0, whence, a fortiori, the equation T = 0 fails in the MV-algebra [O,lJ. 0

2.6. BIBLIOGRAPHICAL REMARKS

2.6

49

Bibliographical remarks

The idea of associating a totally ordered abelian group to any MValgebra A is due to Chang, who in [36] and [38] gave the first purely algebraic proof of the completeness of the Lukasiewicz axioms for the infinite-valued calculus, using quantifier elimination for totally ordered divisible abelian groups. After the unpublished proof of Wajsberg, the literat ure contains many other proofs of the completeness theorem: a proof of Rose and Rosser, based on syntactic methods and linear inequalities [216], the proof in [50] using the representation of free f-groups, the proof of Panti [197] using techniques from algebraic geometry. For the one-variable fragment the reader may also see [193]. Our present geometrie proof, using elementary algebra and convexity theory in finite-dimensional vector spaces, was first published in [51] (also see [52] for its counterpart for lattice-ordered abelian groups). Claim 2 in the proof is a variant of Farkas' lemma (see [250]). Good sequences and the r functor were first introduced in [163].

Chapter 3 Free MV-algebras Free algebras are universal objects: every n-generated MV-algebra Ais a homomorphic image of the free MV-algebra Free n over n generators; if an equation is satisfied by Free n then the equation is automatically satisfied by all MV-algebras. As a consequence of the completeness theorem, Free n is easily described as an MV-algebra of piecewise linear continuous [0, 1]-valued functions defined over the cube [0,1]n. Known as McNaughton functions, they stand to MV-algebras as {O, 1}-valued functions stand to boolean algebras. Many interesting classes of MValgebras can be described as algebras of [0, 1]-valued continuous functions over some compact Hausdorff space. The various representation theorems presented in this chapter all depend on our concrete representation of free MV-algebras.

3.1

McNaughton functions

Let /'i, be an arbitrary, finite or infinite cardinal 2:: 1. (Readers not familiar with ordinals and cardinals may think of /'i, as being a positive integer, without any essential loss.) Suppose we are given distinct propositional variables

for each ordinal a <

/'i,.

Then by definition, each MV-term r in the

51 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

52

CHAPTER 3. FREE MV-ALGEBRAS

variables {Xo}o<1I: is a finite string of symbols over the alphabet

{O, ..." ED, (, ), Xo}o<1I: obtained by the same inductive procedure of Definition l.4.I. For any MV-algebra A the elements of All: have the form

where X o E A for each a < K. The ath projection function 7ro (x) = X O ' We let

Proj~ =def

{7ro

7r0

:

A 11:

~

A is the

la < K}

denote the set of projections of All:. The following is a generalization of Definition 1.4.3:

Definition 3.1.1 For each term T in the variables {Xo}o<1I: the term function TA: All: ~ A is given by induction on the number of connectives in T as follows:

(i)

X;; =def

7ro n

(ii) OA =def the constant function

°over All:,

(iii) (...,p)A =def ...,(pA), (iv) (p ED (T)A =def (pA ED (TA). The dependence of TA on

K

is tacitly understood. We let Term~

denote the set of all term functions over All:. By construction, each element of Term~ is a function only depending on a finite number of variables. Further, Term~ is a subalgebra of the MV-algebra AAl< of all A-valued functions over All:, with pointwise defined operations. More precisely, we have the following result, whose proof immediately follows by definition:

3.1. MCNAUGHTON FUNCTIONS

53

Lemma 3.1.2 For each MV-algebra A and cardinal/'i, 2: 1, Term~ is the smallest subalgebra of AAl< containing each projection 7ra E Proj~.

o

Definition 3.1.3 An MV-algebra A with a distinguished subset Y of elements generating A is said to be free over (the generating set) Y, and is denoted by Freey, iff for every MV-algebra Band every function f : Y --+ B, f can be extended to a homomorphism 1 : A --+ B. Note that 1 is uniquely determined by f. A moment's refiection shows that for any two sets Y and Y' of the same cardinality /'i, , if A is free over Y and A' is free over Y' then A ~ A'. Thus there is no danger of ambiguity in saying that A is the free MV-algebra over /'i, many generators, and writing A = FreeK.. Proposition 3.1.4 For each cardinal/'i, 2: 1, Term~,l] is the free MValgebra over the generating set Projlo,l], in symbols, Term~,l] Proof: Let B be an arbitrary MV-algebra, and f : Projlo,l]

= FreeK..

--+

B be a

function, with the intent of uniquely extending f to a homomorphism --+ B. Let b = (b o, bl , ... ,ba, . . . )a
1 : Term~,l]

if [0,1]1= p = athen B 1= p = a and then pB(b)

= (TB(b).

Stated otherwise, whenever p[O,l] = a[O,l] E Term~·l], then cp maps p and a into the same element of B. Since cp(Xa ) = X:(b) = ba = f(7f a ),

then cp naturally determines an extension 1 of f. By induction one easily sees that 1 is indeed a homomorphism of Term~,l] into B. To prove uniqueness, let 9 : Term~,l] --+ B be a homomorphism extending f. Since 1 and 9 coincide over a subalgebra of AAl< containing all projections, by Lemma 3.1.2 they coincide over all of Term~·l]. 0 The proof of the following proposition is now an immediate consequence of Lemma 1.2.3 and Theorem 1.2.8:

CHAPTER 3. FREE MV-ALGEBRAS

54

Proposition 3.1.5 Let K, ~ 1 be a cardinal. Suppose the MV-algebra A is genera ted by :5 K, elements. Then there is an ideal J in the free MV-algebra FreeK. such that A is isomorphie to the quotient algebra FreeK./ J. 0 Dur next task is to give a more explicit description of the elements

of the algebras FreeK.'

Definition 3.1.6 Let n

~

1 be an integer. Then a function

f : [0, l]n

~

[0, 1]

is called aMeNaughton junetion over [0, l]n iff it satisfies the following conditions: (i) f is continuous with respect to the natural topology of [0, l]n; (ii) there are linear polynomials PI, .. ' ,Pk with integer coefficients,

(bi, mit

E Z), such that for each point Y = (Yo, ... is an index j E {I, ... , k} with 1(Y) = Pj(Y)·

, Yn-l)

E [0,

l]n there

Definition 3.1.7 Let>. be an infinite cardinal. A function g: [0, lJ'~ ~ [0,1] is a McNaughton 1unction over [0, ll~ Hf there are ordinals a(O) < ... < a(m - 1) < >. and a McNaughton function 1 over [0, 1jm such that for each x E [0,1],\ g(x) = f(xo;(o) , ... , Xo;(m-I))'

Proposition 3.1.8 11 ajunction 1 belongs to afree MV-algebra FreeK.' then 1 is a M cNaughton function. Proof: Trivially, the projections are McNaughton functions, and so is the function constantly taking the value Oover [0,1]K.. The set of McNaughton functions is closed under pointwise application of the EB and .., operations. To see that McN aughton functions are closed under the operation EB, if 1 and gare given by linear polynomials PI, ... ,Pm and qll ... , qn, then 1 EB 9 is given by the linear polynomials Pi + qj, (for

3.1. MCNAUGHTON FUNCTIONS

55

all i = 1, ... , m and j = 1, ... , n,) together with the constant function 1. We conclude that the McN aughton functions form a subalgebra of [O,l][O,lJ". By Lemma 3.1.2, all term functions are McNaughton functions. Now apply Proposition 3.1.4. 0 McNaughton's theorem, to be proved in a subsequent chapter, states the converse of Proposition 3.1.8. A short proof of the one-variable case shall be given in the next section. For most applications, however, one does not need the full strength of McNaughton's theorem, but only Lemma 3.1.9 below-a much simpler result. For each real-valued function " we let (3.1)

,tt

=def

b

V 0) /\ 1.

Lemma 3.1.9 Let 9 : [0, l]n --+ R be a linear function with integer coefficients, say, g(x) = moxo+ ... +mn-lxn-l +b, with mo, ... ,mn-I, bE Z. Then gtt E Free n.

Proof: By induction on m = Imol + ... + Imn-ll. If m = 0, then gU coincides with either constant function or 1, whence it trivially belongs to Free n. For the induction step, assume the lemma holds for m - 1. It is no loss of generality to assume that max(lmol, ... , Imn-ll) = Imol. Proceeding by cases, assume first mo > 0. Let h = 9 - Xo, so that

°

(3.2) h = h(xo, ... , Xn-l) = b + (mo - l)xo

+ mlXl + ... + mn-lXn-l·

By induction hypothesis, both functions h t and (h+1)tt belong to Free n. We shall prove that for each x = (xo, . .. , Xn-l) E [O,l]n,

The identity trivially holds whenever x is such that h(x) > 1, or h(x) < -1. If x is such that h(x) E [0,1], then htt(x) = h(x), and (h(x) + l)U = 1. Since Xo E [0,1], (h(x) + xo)tt = h(x) EB Xo, which establishes (3.3). Finally, if h(x) E [-1,0}, then M(x) = and (h(x) + l)U = h(x) + 1, whence (3.3) follows from the identities (h(x)+xo)U = max(O, h(x)+xo) = max(O, Xo + h(x) + 1 - 1) = x00 (h(x) + 1). Thus, in case mo > 0,

°

CHAPTER 3. FREE MV-ALGEBRAS

56

identity (3.3) holds for each x = (xo, ... , Xn-l). By induction, together with Proposition 3.1.4, we have (h+xo)~ = g~ E Free n . In case mo < 0, applying the same argument to the function 1 - g, one shows that (1 - g)~ E Freen- Since 1 - (1 - g)~ =g~, we conclude that gtt E Free n in all possible cases. 0

3.2

The one-dimensional case

In this section we shall prove that, up to isomorphism, the free MValgebra Freel over one generator coincides with the MV-algebra of one-variable McNaughton functions. To this purpose, for each n = 0, 1, ... , the nth Farey partition FareYn of the unit interval [0,1] is defined by Fareyo =def

{0,1},

FareYl =def {O,

1

2' 1}, 112

FareY2 =def {O, 3'

FareY4 =def

2' 3' 1},

1 121 323 {O, 4' 3' 5' 2' 5' 3' 4' I}, 112 132 3 1 4 3 5 2 5 3 4

{0'5'4'7'3'8'5'7'2'7'5'8'3'7'4'5,1},

Thus, FareYn+1 is obtained by inserting between any two consecutive elements alb and eid of FareYn their mediant (a + e)/(b + d), with = 0ll and 1 = 1/1. Dur present FareYn is a notationally simpler variant of the traditional "Farey sequence" of order n + 1, where one inserts only those mediants whose denominators do not exceed the value n + 1. Farey observed, and Cauchy proved, the following elementary properties of Farey partitions:

°

Proposition 3.2.1 Let n = 0,1,2, .... Then we have (i) All jractions in FareYn are automatieally in irredueible form, and the interval [alb, eid] determined by any two eonseeutive fraetions alb< eid in FareYn has the unimodularity property eb - ad = 1. Moreover, alb< (a + e)/(b + d) < eid.

3.2. THE ONE-DIMENSIONAL GASE

57

(ii) Every irreducible fraction p/q E [O,IJ occurs in FareYm, for some index m. Proof: (i) Let (1,1) and (0,1) be the vectors in Z2 respectively giving the homogeneous correspondents of 1 and 0. Then the fact that the closed interval [O/I,I/IJ satisfies unimodularity has the following equivalent reformulations: (a) the determinant of the matrix whose rows are given by (1,1) and (0,1) is equal to one; (b) every vector in Z2 is a linear combination, with integer coefficients, ofthe vectors (1, 1) and (0, 1)-for short, the pair ((1, 1), (0, 1)) is a basis in Z2. One then immediately sees that if an interval [u/v,u'/v'J ~ [O,IJ satisfies the uni modular law, then so do the two intervals ru/v, u" /v"J and [u"/v",u'/v'J where u"/v" =def (u + u')/(v + v'). For short, unimodularity is preserved under the operation of taking mediants. The rest is clear. To prove (ii), by way of contradiction assurne < p/q < 1 to be an irreducible fraction not occurring in any FareYn- The coordinates of vector (p, q) E Z2 in the basis B o =def ((1,1), (0, 1)) are given by p and q - p. Note that their sum is strictly less than the sum p + q of the coordinates of the same vector in the initial basis ((1,0), (0, 1)). Taking the mediant 1/2 of 0/1 and 1/1 (the latter two fractions giving the inhomogeneous correspondents of and 1), and passing to homogeneous coordinates in Z2, we obtain two new bases ((1,1), (1,2)) and ((1,2), (0, 1)). Precisely one of them, denoted BI, encapsulates the vector (p, q), in the sense that the coordinates al and ßl of (p, q) with respect to basis BI are integers 2: 1. By direct inspection, the sum of these coordinates is strictly less than q. Proceeding inductively, assurne we are given an encapsulating basis B j = (Vj, Wj), for suitable integer vectors Vj and Wj in Z2, and let aj and ßj be the coordinates of (p, q) with respect to B j . Upon taking the mediant of the inhomogeneous correspondents of Vj and Wj, we obtain two bases (Vj, Vj + Wj) and (Vj + Wj' Wj); precisely one of them, denoted Bj+I, encapsulates (p, q); further, the coordinates aj+l and ßj+l of (p, q) with respect to B Hl satisfy the inequality 2 :::; aj+l + ßj+l < aj + ßj. Thus, by our absurdum hypothesis, the sum of the coordinates of (p, q) decreases infinitely

°

°

CHAPTER 3. FREE MV-ALGEBRAS

58

often, which is impossible. 0 The 2n + 1 elements of Fareyn can be displayed in increasing order as follows: o < Q < ... < I < 8 = eid< c < ... < w < 1. A (Schauder) hat of FareYn is a function of either form:

lid

w

E

Thus each Schauder hat is a continuous piecewise linear function h : [0, 1] -+ [0, 1), whose graph consists of the four segments joining the points (0,0), (f,0), (8, lid), (c,O) and (1,0). As a consequence of unimodularity, each linear piece of h has the form y = mx + q, for suitable integers m and q. Of course, the graphs of the two extrem al hats only consist of two segments. For each n = 0,1,2, ... , we shall denote by Schaudern =def (h l , ... ,hu ) the naturally ordered sequence of all Schauder hats of FareYn, where u = 2n + 1. We naturally say that hj and h j +1 are contiguous. The following is a picture of Schauderi, for i = 0,1,2:

Schaudero

Schauderl

Schauder2

3.2. THE ONE-DIMENSIONAL GASE

59

For each hat hi of Schaudern the multiplicity J.-li of hi is the inverse of the maximum value of h i . Thus by definition, J.-li coincides with the denominator di of the rational point Ci/ di E [0, 1] at which hi attains its maximum value. For any n, the Schauder hat hi precedes hj in Schaudern iff i < j iff ci/di < cj/dj . Note that if li-jl ::::: 2 then hil\hj = O. From the sequence (h 1 , ••. , hu ) one can unambiguously recover the multiplicity of each hat, as wen as the number n = log2(u - 1). It is easy to see that Ei J.-lihi = 1. By anode of a continuous function / : [0, 1] ---+ [0, 1] we mean a nondifferentiability point in the domain of /. The vertices and 1 are always included among the no des of /. If / arises from the identity function via a finite number of applications of the operations ..." EB, 8, 1\ and V, then the no des of / are finitely many, and they are an rational numbers. By Proposition 3.1.4 this is always the case when / E Freel.

°

Proposition 3.2.2 Let h 1 , ... , h u be the hats 0/ Schaudern, in their natural order, and with their respective multiplicities J.-l1, ... , 11-1.1. Let k1 , ... , k2u - 1 be the hats 0/ Schaudern+l' with their respective multiplicities 6, ... , 61.1-1. Then we have:

k2i = hi

1\ hi+1!

with 6i = (l1-i + 11-i+1) , /or each i = 1,2, ... , u - 1;

k2i - 1 = hi - (h i 1\ (h i- 1 V hi+1)) = hi - (h i 1\ (h i - 1 + hi+l)) hi e (h i- 1 EB hi+d, with 6i-1 = J.-li, /or each i = 2,3, ... , u - 1.

°

=

Prao!" Let = cd d 1 < C2/ d2 < ... < cu/du = 1 be the ascending sequence of elements of FareYn. Then di = J.-li. Direct inspection, in the light of the unimodularity property in Proposition 3.2.1 (i), shows that the function hi 1\ hi+1 attains its maximum value 1/(di + di + 1 ) at the mediant point p = (Ci + ci+l)/(di + di+d. Since the nodes of h i 1\ hi+1 are given by the five rational numbers 0, ci/di , p, Ci+l/di+1, 1, and the function coincides with k 2i at each node, then k2i coincides with hi 1\ hi+l over an of [0, 1]. Similarly, the function hi - (h i 1\ (h i - 1+ hi+l)) is constantly equal to zero over both intervals [0, (Ci-l + ci)/(di- 1 +

60

CHAPTER 3. FREE MV-ALGEBRAS

di )] and [(CiH + Ci)/(di+l + di ), 1], and is linear over both intervals [(Ci-l + ci)/(di- l + di ), Ci/di] and [Ci/di , (Ci+l + Ci)/(di+l + di )]. Since (hi - l A hi+l) = 0, the function coincides with k 2i - 1 at the point cddi , where it attains its maximum value 1/di . It is now easy to see that the identity k 2i - l = h i - (h i A (h i- l + h iH )) holds over all of [0,1]. One

similarly proves the remaining identities. 0 By direct inspection we have

Proposition 3.2.3 (i) Each one of the extremal hats h l and h u of Schaudern is the sum of two hats of Schaudern+l; all the remaining hats of Schaudern are obtainable as a sum of three hats of Schaudern+l. (ii) Suppose a linear combination Alh l + ... + Auhu with real coefficients Ai coincides with the constant function 0. Then each coefficient Ai must vanish. For short, the hats of Schaudern form a linearly independent set of functions. 0

Definition 3.2.4 Fix an integer n = 0,1, ... , and let h l , ... , hu be the hats of Schaudern, with their respective multiplicities Ih, ... , f.Lu. Then by a subsystem S of Schaudern (in symbols, and with a slight abuse of notation, S ~ Schaudern) we mean a sequence of integers Al, ... , Au, with Ai :::; f.Li for each i = 1, ... , u. The associated function fs : [0, 1] ~ [0,1] is defined by fs = Ei Aihi. We further let S-' ~ Schaudern =def (f.Ll - Al,· .. , f.Lu - Au), whence, recalling that f.Llhl + ... + f.Luhu = 1, we have the identity fs~ = -,fs = 1 - fs. Two subsystems S ~ Schaudern and R ~ Schauderm are said to be equivalent iff fR = f s·

°: :;

From Proposition 3.2.3 we get ~ Schaudern then for each integer p 2:: n there is precisely one subsystem S' ~ Schauderp equivalent to S. 0

Proposition 3.2.5 1f S

Proposition 3.2.6 Given two subsystems R ~ Schauder m and S ~ Schaudern, there exists an integer q 2:: max(m, n) and a subsystem T ~ Schauder q such that fT

=

fR ffi fs.

Proof: Let q be the smallest integer 2:: max( m, n) such that all the nodes ofthe function fRffifs = min(1, fR+ fs) occur in FareYq. The existence

3.2. THE ONE-DIMENSIONAL GASE

61

of q follows from Proposition 3.2.1 (ii), together with Proposition 3.2.2 ensuring that all the no des of IRffJ!s are rational numbers. Let 9 = IR+ Is. For any two consecutive fractions x and y in FareYq one cannot have g(x) > 1 and g(y) < 1. Let R' = (..\1"'" ..\w) and S' = (111,"" IIw) be the equivalent subsystems of S and of R in Schauderq, as given by Proposition 3.2.5. Let kI , ... , k w be the hats in Schauderq, with their multiplicities 6, ... , ~w' Thus, for any two consecutive indexes i, j E {1, ... , w}, if ..\j + IIj < ~j then necessarily ..\i + lIi ~ ~i' Let T ~ Schauderq be the subsystem ('f/I, ... , 'f/w), where 'f/i = min(~i,..\i + IId. It follows that the function IT coincides with IR EB Is at each point of FareYq, and both functions are linear on each interval between consecutive points of FareYq. Thus, IT = IR EB Is as required. 0

°

Example: With the notation of Proposition 3.2.6, let m = n = and R = S = (0,1), so that Schauderm = (1 - x, x) and IR = Is = X. The nodes of the function IR EB Is = min(2x, 1) are 0, 1/2 and 1, which all occur in the Farey partition FareYq for q = 1. By definition,

Schauder 1

= (max(l -

2x, 0), min(x,l - x), max(O, 2x - 1)),

with multiplicities 1,2,1, respectively. As a particular case of Proposition 3.2.5, the subsystem R' ~ Schauderi given by the triplet (0,1,1) is equivalent to R. Indeed, IR = IRI = x. Letting the subsystem T of Schauderi be defined by T = (0,2,1), we have IR EB Is = Ir. For no subsystem U of Schaudero we would be able to write IR EB Is = lu· Theorem 3.2.7 Let I : [0,1] - [0,1] be an arbitrary lunction. Then the lollowing are equivalent: (i) I arises from the identity function x : [0, 1] - [0, 1] via a finite number 01 applications 01 the operations --, and EB; (ii) there is a subsystem T such that I = IT.

Proof" One direction immediately follows from Proposition 3.2.2, since both operations 0 and A are definable in terms of --, and EB. For the other direction, as shown by the above example, the identity function x arises as IR for some subsystem R. Proceeding by induction on the number of operation symbols --, and EB needed to obtain I, in case I = --,g one simply not es that Is~ = --,(Is). In the remaining case,

62

CHAPTER 3. FREE MV-ALGEBRAS

from Propositions 3.2.5 and 3.2.6 it follows that the set of functions of the form Ir, for T a subsystem T of some Schaudern is closed under applications of -, and EB. This yields the desired conclusion. 0

Corollary 3.2.8 The free MV-algebra mel over one generator is isomorphie to the MV-algebra 0/ McNaughton junctions %ne variable. Proof: By Proposition 3.1.4, Freel is a subalgebra of the MV-algebra of one-variable McNaughton functions. Conversely, given any such function /, let us display its nodes as follows:

o= ni < n2 < ... < nu-I< n u = 1. As already remarked, every ni is a rational number, and / is linear over each interval [ni, ni+1]' By Proposition 3.2.1(ii), there exists an integer q such that all ni's are members of FareYq. Let

o=

ml < m2 < ... <

mv-l

<

m v

= 1

display the elements of FareYq. Note that the value of / at each node mj = aj/bj is a multiple of l/bj , for some integer 0 ~ t ~ bj . It follows that a suitable linear combination g (with integer coefficients) of the hats in Schauderq will coincide with / at all nodes of mj. Since both g and / are linear over each interval [mi, mi+l], then g and / will coincide over the interval [0,1]. There remains to be shown that every Schauder hat is an element of Freel' This is an immediate consequence of Theorem 3.2.7 and Proposition 3.1.4. 0

3.3

Decomposing McNaughton functions

In this section, for later use, we discuss so me properties of McN aughton functions of n variables. Let / : [0, l]n - [0,1] be a McNaughton function. Recalling Definition 3.1.6, let PI, ... ,Pk be the linear eonstituents of /, i.e., the distinct polynomials representing the linear pieces of /. For any two such polynomials P and q, either P ~ q, or q ~ p, or there is a hyperplane H dividing Rn into two closed half-spaces H+ and H- such that p(x) ~ q(x)

3.3. DECOMPOSING MCNAUGHTON FUNCTIONS

63

holds for each point x E H+, while q(x) ~ p(x) holds for each point x E H-. More generally, for every permutation p of the set {I, ... , k}, let Pp ~ [0, l]n be defined by

(3.4)

Pp =def

{x E [0, l]n

I Pp(l) (x) ~ Pp(2) (x)

~

... ~ Pp(k) (x)}.

As an intersection of the cube [0, l]n with a finite set of closed halfspaces, each Pp is, by definition, a (possibly empty) convex compact polyhedron. By definition, the vertices of Pp are those points of Pp that cannot be expressed as nontrivial convex combinations of points of Pp. By the fundamental theorem on polyhedra, Pp can be visualized as the convex hull of the finite set of its vertices. Since all linear constituents of f have integer coefficients, each vertex v of Pp is rational, in the sense that the coordinates of v are rational numbers. Let us agree to denote by C the set of n-dimensional polyhedra of the form Pp, for some permutation p, in the above decomposition of the domain of f. Then C is a finite set of compact convex n-dimensional polyhedra with rational vertices, having the following additional properties: (i) the union of all polyhedra in C coincides with the cube [0, l]n, (ii) any two polyhedra in C are either disjoint or they intersect in a common face, and (iii) for each polyhedron P E C there is an index u = Up E {I, ... , k} such that upon restriction to P, the two functions fand Pu. coincide. For furt her applications, it will be convenient to replace C with a family of n-dimensional simplexes satisfying (i)-(iii). To this purpose, one can simply triangulate every polyhedron in C, generalizing the familiar triangulation of convex polygons in the two-dimensional case. In this way one does not even need to add new vertices. As an alternative construction, for each d = 0, ... ,n let us denote by :;:(d) the set of d-dimensional faces of polyhedra in C. Thus, for instance, :;:(0) is the set of singletons given by the vertices of polyhedra in C, :;:(1) is the set of edges of polyhedra in C, ... , :;:(n) = C. For every j = 2,3, ... , n and every polyhedron P E :;:(j) let us select, once and for all , a rational point b p in the relative interior of P. In other words, b pEP n Qn and b p does not belong to any

CHAPTER 3. FREE MV-ALGEBRAS

64

(j - 1)-dimensional face of P. We now inductively define a sequence S(O) , S(I) , ... ,s(n) where each SO) is a set of j-dimensional simplexes, as folIows: S(O)

= ;:(0) , S(I) =

S(Hl) = {[bp, F]

;:(1)

,

I PE ;:(Hl) , FE S(i) , F

~ P},

where [bp, F] is the simplex whose vertices are the vertices of F together with b p . Letting S = s(n), we have proved:

Proposition 3.3.1 For any McNaughton function f : [O,l]n - [0,1] with its linear constituents PI, ... ,Pk, there is a set S of compact ndimensional simplexes with rational vertices such that (i) The union of all simplexes in S coincides with the cube [0, l]n ,.

(ii) Any two simplexes in S are either disjoint or they intersect in a common face; (iii) For each simplex WES there is an index u E {I, ... , k} such that, upon restriction to W, the two functions fand Pu coincide.

o

3.4

Ideals in free MV-algebras

The aim of this section is to describe the maximal ideals in free MValgebras and to give conditions ensuring that an ideal of a free MValgebra is an intersection of maximal ideals. Dur results essentially depend on the fact that free MV-algebras are algebras of [0, l]-valued continuous functions on a compact Hausdorff topological space. Therefore, we shall firstly recall the ideal theory of such function algebras.

3.4. IDEALS IN FREE MV-ALGEBRAS

65

As the reader will remember, for every MV-algebra A, T(A) and M(A) respectively denote the set of ideals and of maximal ideals of A. For X an arbitrary nonempty set, let A be a subalgebra of the MV-algebra [0, 1]X of all functions f: X --+ [0,1], with pointwise defined operations. For each set S ~ X, let (3.5)

Js

=def {f E

AI f(x) =

°for all x ES},

be the set of functions vanishing over S. Further, for any ideal J in A let

(3.6)

VJ =def n{f- 1 (0)

1/ E J}

be the intersection of the zerosets Z(J) =def f- 1 (0) of all functions f E J. Trivially, Js E T(A); further, J0 = A and Jx = {O} = (0). For each x E X we shall write J x instead of J{x}.

Lemma 3.4.1 Let X be a nonempty set and let A be a subalgebra of the MV-algebra [0, 1]x. Thenfor each x E X, Jx E M(A). Proof: Suppose first that A = [0,1]x. Jx is a proper ideal of A, because the constant function 1 is not among its elements. If f E A \ Jx , then f(x) > 0, and we can find an integer n ~ 1 such that nf(x) 2:: 1. It follows that ->nf = 1 - (J ffi ... ffi f) E Jx , ,

n

'" times

"

whence, by Proposition 1.2.2, Jx E M(A). Now to complete the proof it suffices to apply Proposition 1.2.16 (ii). 0 For any topological space X we shall denote by Cont(X)

the subalgebra of [0,1]X given by the continuous [0,1]-valued functions over X. It is understood that [0, 1] is equipped with its natural topology.

Proposition 3.4.2 Let X be a compact Hausdor.fJ space, and A be a subalgebra of the MV-algebra Cont(X). The map J ~ V J is an inclusion reversing map from the set T(A) of ideals into the family of closed subsets of X. Moreover, VJ =f 0 for each proper ideal J in A.

66

CHAPTER 3. FREE MV-ALGEBRAS

Proof" The continuity of each function I E A ensures that VJ is a closed subset of X; trivially, the map J 1-+ VJ reverses inclusions. Let J be a proper ideal in A. By way of contradiction, suppose VJ = 0. Then by the assumed compactness of X there are finitely many functions 11, ... , Is E J such that the intersection of their zerosets is empty. Let I = !I E9 ... E9 Is· Then I E J and the zeroset of I is empty. Since I attains a minimum value > 0, there exists an integer m ~ 1 such that m/(x) > 1 for all x E X. We conclude that I E9 ... E9 I (m times), constantly takes the value 1. Thus 1 E J, and J = A, a contradiction.

o

A subalgebra A of [0, 1jX is said to be separating iff for any two distinct points x and Y in X, there is I E A such that I (x) = 0 and

I(y) > o.

Theorem 3.4.3 Let X be a compact Hausdorff space and A be a separating subalgebra 01 Cont(X). Then we have

(i) The map x M(A);

1-+

Jx is a one-one correspondence between X and

(ii) For each closed set S

~

X, VJs = S;

(iii) For each proper ideal J in A, JVJ is the intersection 01 alt maximal ideals in A containing J. Proof" (i) By Lemma 3.4.1 together with our assumption about A, the map x 1-+ Jx is a one-one correspondence from X into M(A). To see that this map is onto M(A), let J be a maximal ideal of A. By Proposition 3.4.2, VJ is a nonempty closed subset of X. Since for each y E VJ , J y 2 J, we conclude that VJ is a singleton, say VJ = {x} and J= Jx ' (ii) Trivially, S ~ VJs ' To prove the converse inclusion, suppose Z E X \ S. Since A is separating, for each y E S we can find I y E A such that Iy(z) = 0 and ly(Y) = ay > O. By continuity, there is an open neighborhood Uy of Y such that Iy(x) > by = ay/2 for each x E Uy. By a standard compactness argument, there are finitely many functions 11, ... ,!k E A, together with real numbers numbers b1, ... ,bk > 0 such that, upon defining I = !I E9 ... E9 Ik, we can write I(z) = 0 and

3.4. IDEALS IN FREE MV-ALGEBRAS

67

°

f(x) > min(b1 , ... , bk ) > for each x E S. Hence for some integer n ~ 1, we must have -.nf E J s and -.nf(z) = 1, thus showing that z rt VJs · This yields the desired conclusion VJs ~ S. (iii) For each S ~ X, Js = nXES Jx . On the other hand, J ~ Jx if and only if xE VJ . The desired result now follows from (i). 0 Remark: Let X be a compact Hausdorff space and A be a separating subalgebra of Cont(X). It follows from (ii) in the above theorem that each closed subset of X is an intersection of zerosets of functions in A. Therefore, the complements of the zerosets of functions in A form an open basis for the topology of X. Corollary 3.4.4 Let X be a compact Hausdorff space, let A be a separating subalgebra of Cont(X), and J be an ideal in A. Then J is an intersection of maximal ideals iff J = JVr Moreover, the map U J--+ J X\U is an order isomorphism from the set of proper open subsets of X and the set of ideals of A that are intersection of maximal ideals (both sets being ordered by inclusion). The inverse isomorphism is given by J J--+ X \ VJ . 0 Example: Let f: [0, 1]-+ [0,1] be defined by f(x) = x sin(l/x), for = 0; let further f+(x) = max(f(x) , 0) for each x E [0,1]. Then (1+) is a proper ideal of Cont([O, 1]) and it is not hard to verify that JV(f+) strictly contains (1+). Therefore, (1+) is a principal proper ideal of Cont([O, 1]) which is not an intersection of maximal ideals.

°< x ::; 1 and f(O)

Let X be a compact Hausdorff space, and A a sub algebra of the MV-algebra [O,I]x. For every nonempty subset Y of X, let

denote the subalgebra of [O,l]Y given by the restrietions to Y of the functions in A. The map f J--+ pU) = fly defines a surjective homomorphism p: A -+ Aly. In case Y happens to coincide with VJ for some proper ideal J in A, it follows that Ker(p) = JVr Then from Lemma 1.2.7 we immediately obtain

68

CHAPTER 3. FREE MV-ALGEBRAS

Proposition 3.4.5 Let X be a compact HausdorfJ space, and A be a separating subalgebra 01 Cont(X). Por each J E I(A), the map I I J ~ IlvJ is an isomorphism from AI J onto AlvJ il and only il J is an intersection 01 maximal ideals 01 A. 0

In order to apply the above results to free MV-algebras, we prepare Lemma 3.4.6 For each cardinal/'i, 2:: 1, Free", is a separating subalgebra 01 Cont([O, 1]"'). Proof" In the light of Proposition 3.1.8, we may visualize the elements of Free", as McNaughton functions on the cube [0,1]"', the latter being equipped with product topology. Therefore, Free", is a subalgebra of Cont([O, 1]"'). In order to show that this sub algebra is separating, let Y = (Yo, Yl, ... ) and z = (zo, Zl,' .. ) be two distinct points of [0,1]"'. Assume without loss of generality Yo < Zoo Let r be a rational number such that Yo < r < Zoo Let p = p(x) = ax + b be a linear polynomial with integer coefficients such that a > 0 and r = -bla. Then by Proposition 3.1.4 and Lemma 3.1.9, the McNaughton function I(x) = p#(xo) belongs to the free MV-algebra Free"" and, moreover, I(y) = 0 and I(z) > o. 0

As an immediate consequence of the above lemma together with Theorem 3.4.3, we record here the following result, that will find application in subsequent sections. Proposition 3.4.7 The map x ~ J x is a one-one correspondence between points 01 [0, 1]'" and maximal ideals 01 Free",. The inverse correspondence is given by JE M(Free",) ~ the only point oIVJ . 0

The following lemma is of independent interest: Lemma 3.4.8 Let I, 9 E Free",. Then (3.7)

gE(J)

ifJ Zg2ZI·

3.4. IDEALS IN FREE MV-ALGEBRAS Proof: For the nontrivial direction, assuming Z 9

69

2

Z f, since both 9

and f only depend on finitely many variables, say Xl,"" X m we can safely restrict attention to [0, l]n. By a simple adaptation of Proposition 3.3.1 there exists a set S = {Tb"" Tu} of compact convex n-dimensional simplexes, whose union coincides with [0, l]n, any two Ti and T; being either disjoint or intersecting in a common face, and with the additional property that both functions fand 9 are linear over each Ti. Let ViO,"" Vin be the vertices of the simplex Ti. For each 1 ~ i ~ u and ~ j ~ n there is an integer mij ~ 1 such that g(Vij) ~ mijf(vij). As a matter of fact, if f(Vij) = then by hypothesis g(Vij) = 0, and we can take mij = 1. If, on the other hand, f(Vij) > 0, then the existence of mij follows from the archimedean property of the real numbers. Let mi = max(miQ, ... , min). Since each x E Ti is a convex combination of the vertices of Ti, and since both 9 and f are linear over 7i, we get g(x) ~ md(x) for each xE 7i. Let m = max(mb"" m u ). From [O,l]n = Uf=l7i , it follows that g(x) ~ mf(x) for each x E [O,l]n. Since 9 ~ 1, we finally obtain g(x) ~ min(l, mf(x)) = f(x) $ ... $ f(x) (m times). In conclusion, gE (1). 0

°

°

In contrast with the example given after Corollary 3.4.4, we have Theorem 3.4.9 Each proper principal ideal of FreeK, is an intersection of maximal ideals. Proof: An immediate consequence of Corollary 3.4.4, Lemma 3.4.6,

together with Lemma 3.4.8. 0 In the next section we shall give an example of a nonprincipal ideal of Freel which is an intersection of maximal ideals. The following example shows that free MV-algebras also contain proper ideals which are not intersections of maximal ideals. Example: With reference to formula (3.1), for each k = 1,2, ... , let the

McNaughton function

!k : [0,1] - [0,1] be defined by fk(X) = (k - (k + l)x)~.

Let J be the ideal of Freel generated by the functions JI, 12, .... By direct inspection we easily obtain VJ = {1}, and (1 - x)~ E J l \ J. Therefore, J =f JVr

70

3.5

CHAPTER 3. FREE MV-ALGEBRAS

Simple MV-algebras

An MV-algebra is called simple Hf it has exact1y two ideals. In other words, an MV-algebra Ais simple if and only if Ais nontrivial and {O} is its only proper ideal. Theorem 3.5.1 For every MV-algebra A the following eonditions are equivalent:

(i) A is simple; (ii) A is nontrivial and for every norizero element x E A there is an integer n > 0 sueh that 1 = x EB " . EB x (n times);

(iii) A is isomorphie to a subalgebra of [0,1]. Proof: In the light of Proposition 1.2.2, (ii) states that {O} is a maximal ideal of A. Therefore, (i) and (ii) are equivalent. It is obvious that (ii) is satisfied by all subalgebras of the MV-algebra [0,1]. Finally, to prove (i) -+ (iii), assume A to be simple. If the cardinality of A is K" then by Proposition 3.1.5, we can identify A with the quotient MValgebra Free"./ J for some ideal J of Free lt • By Proposition 1.2.10, since A is simple, J must be a maximal ideal of Free lt • Therefore, by Proposition 3.4.7, J = Jx , for a uniquely determined point x E [0,1]1t. Applying now Proposition 3.4.5, we obtain that A is isomorphie to the MV-algebra Freeltl{x} = 7fx (Free lt ), where 7rx: Free lt -+ [0,1] is the projection given by 7rx (f) = f(x). Hence A is isomorphie to a subalgebra of [0,1].0 Corollary 3.5.2 Every simple MV algebra A has at most the eardinality of the eontinuum. 0 To obtain further information about simple MV-algebras, recall that for each integer n = 2,3, ... , the n element (Lukasiewicz) chain Ln is defined by 1 2 n- 2 (3.8) Ln =def {O, - - , - - , ... , - - , I}. n-1 n-1 n-1

3.5. SIMPLE MV-ALGEBRAS

71

Proposition 3.5.3 Let A be a subalgebra of [0,1]. Let A+ = {x E A I x > O}, and a = inf A+ be the infimum of A+. 1f a = then A is a dense subehain of [0, 1]. 1f a > then A = Ln for some n ;::: 2.

° °

°

°

°

Proof: In case a = 0, let < z ::; 1 and < € < z/2. By assumption there is b E A + such that < b < €. Letting n be the smallest integer such that nb;::: z, noting that n > 2, it follows that Z-€ < (n-l)b < z. Thus A is dense. In case a > 0, if a = 1 then A = L 2 = {O, I}. If a < 1, since A is closed under the operation 1 - x, we have a::; 1/2 and a E A+. For otherwise, there exist two elements x, y E A+ such that a < x < y < 2a, whence a > y - x = y 0 -,x E A+, a contradiction. Having proved that a E A +, let m be the unique integer such that (m - l)a < 1 ::; ma. Note that m ;::: 2. Let K

= {O, a, 2a, ... , (m - l)a, I}

~

A.

We shall prove that K = A. For otherwise, let x E A\K (absurdum hypothesis ). If (m - l)a < x < 1 then -,x would be a nonzero element of A strictly smaller than a, which is impossible. If for some j = O,I, ... ,m - 2 we have ja < x < (j + l)a then, again, (j + l)ax = (j + l)a 0 -,x is a nonzero element of A which is strictly smaller than a, another contradiction. Having thus proved that K = A, since < a < 2a < '" < (m - l)a < 1, it follows that a = 1 - (m - 1 )a, whence a = l/m and A = L(m+l), as required. 0

°

Remark: Let m, n ;::: 2. Then from the above proof it follows that L m ~ Ln iff for some k E {I, ... , n -I} we have the identity m~l = n~l iff m - 1 is a divisor of n - 1.

Corollary 3.5.4 An MV-algebra A is finite and simple if and only if A is isomorphie to an MV-algebra Ln for some integer n ;::: 2. Proof: For the nontrivial direction, if A is finite and simple, by Theorem 3.5.1 we can identify A with a finite sub algebra B of [0,1]. By Proposition 3.5.3, letting n ;::: 2 be the number of elements of B, we have that B = Ln. 0

As promised, we shall now show that there are maximal ideals in Freel that are not principal.

CHAPTER 3. FREE MV-ALGEBRAS

72

Example: Consider the subalgebra BA of [0,1] generated by an irrational number A E [0,1]. Sinee BA is simple and has one generator, we ean write BA = Freed J for some maximal ideal J of Freel' Now, J eannot be a principal ideal: for otherwise, (absurdum hypothesis ), reealling Proposition 3.4.7 and writing J = Jx = (J), for some x E [0,1], and 1 E Freel, we get that the zeroset Z 1 eoincides with the singleton {x}. On the otherhand, sinee all linear pieees of 1 have integer eoeffieients, x must be a rational number, say x = alb for relatively prime integers ~ a ~ b with b > 0. It follows that Freed J is isomorphie to the MV-algebra Lb+l of possible values of MeNaughton functions g at the rational point alb. Sinee by Proposition 3.5.3, BA is infinite and L b+ 1 is finite, these two algebras are not isomorphie. Thus J is nonprineipal.

°

3.6

Semisimple MV-algebras

In the light of Corollary 1.2.15, for any MV-algebra A, we eall radical 01 A the interseetion of all maximal ideals of A. The radieal of A will be denoted by Rad(A). An MV-algebra A is said to be semisimple Hf A is nontrivial and Rad(A) = {O}. In partieular, every simple MV-algebra is semisimple. In what follows, we will eonsider only nontrivial MV-algebras. As an immediate eonsequenee of Proposition 1.2.10, for each ideal J of A, the quotient AI J is a simple MV-algebra if and only if J is maximal. Henee, by Theorem 3.5.1, AI J is isomorphie to a subalgebra of [0, 1] if and only if J is a maximal ideal of A. The next proposition will be promptly reeognized as an immediate eonsequenee of Theorem 1.3.2: Proposition 3.6.1 An MV-algebra is semisimple il and only il it is a subdirect product 01 subalgebras 01 [0,1]. 0 Corollary 3.6.2 Every free MV-algebra is semisimple. Proof: Immediate from Proposition 3.1.4. 0

Our next aim is to eharaeterize the elements of the radical.

3.6. SEMISIMPLE MV-ALGEBRAS

73

Definition 3.6.3 An element a in an MV-algebra A is said to be infinitely small or infinitesimal iff a =I- 0 and na ~ ,a for each integer n ;::: O. The set of all infinitesimals in A will be denoted by Infinit(A). Remark: If ais infinitesimal, then the elements na, for n = 0,1,2, ... , form a strictly increasing sequence. Indeed, if for some n, na = (n+ 1 )a, then one would have 1 = ,((n + l)a) EB na = ,a V na = ,a, i.e. a = 0, a contradiction. Proposition 3.6.4 For any MV-algebra A, Rad(A)

{O}.

= Infinit(A) U

Proof: Suppose a rt Rad(A). Then there is a maximal ideal of A, say M, such that a rt M. Hence (M U {al), the ideal generated by a and M, must coincide with A, and by (1.9), there is an integer n 2: 0 and an element z E M such that 1 = na EB z. If na ~ ,a, then one would have a ~ -.na ~ z, whence a E M, a contradiction. Hence a is not infinitesimal, whence Infinit(A) ~ Rad(A). Conversely, assurne that 0 < a E A is not infinitesimal. Then there is an integer m ;::: 0 such that ma 1:. -.a. Hence ma e -.a = ma 0 a > 0, and by Proposition 1.2.13, there is a prime ideal P of A such that ma e -.a rt P. By Equation (1.7) we have ,a e ma E P. By Corollary 1.2.12 there is a maximal ideal M of A such that P ~ M. Then -.(aEBma) = ,aema E M. Therefore, (m+1)a rt M, and hence, art M. In conclusion, art Rad(A), and Rad(A) ~ Infinit(A) U {O}, as required to complete the proof. 0 It follows from the above proposition that an MV-algebra A is semisimple if and only if A has no infinitesimals. In particular, by the remark following Definition 3.6.3, each finite MV-algebra Ais semisimpIe; moreover, since A has only a finite number of ideals, A is the subdirect product of a finite family of finite subalgebras of [0, 1]. The next proposition gives a sharper result:

Proposition 3.6.5 An MV-algebra A is finite if and only if A is isomorphie to a finite produet of finite ehains, in symbols, (3.9) A

~

Ld1

X •.• X

Ld,., for some integers 2 ::; d 1

::;

d2

~ ••• ~

This representation is unique, up to the ordering of faetors.

du.

74

CHAPTER 3. FREE MV-ALGEBRAS

Proof: For the nontrivial direction, assume A is a finite MV-algebra. As noted above, Ais semisimple. Let 11 , ••• ,Iv. be the list of an distinct maximal ideals of A. Then A is a subdirect product of the MV-algebras AlI1 , AII2 , ••• , AI Iv.. By Corollary 3.5.4, for each t = 1, ... , u the quotient algebra AlIt can be identified with the chain L dt , for some integer dt 2: 2. Let us now define the produet MV-algebra P by

P

= II{L dt I t = 1, ... , u}.

For each element x E A and every ideal I t the element x I I t E L dt ean be identified with a rational number cl (dt - 1), for some integer o ~ c ~ dt -1. Let ß: x E A ~ (xIIll ... ,xIIv.) E P. Then from the semisimplicity of A it follows that ß is injeetive. To see that ß is surjeetive, it is sufficient to show that every element in P of the form (0, ... , 1/(dt - 1), ... ,0) is in the range of ß. To this purpose, let at be the minimum of an elements z E A such that z I I t = 1I (d t - 1). The existence of at is ensured by the assumed finiteness of A. For every maximal ideal Ir of A other than I t , there is an element w E A such that at ~ wand w I Ir = O. For otherwise, Ir ~ I t , thus contradicting the maximality of Ir. We have proved that ß(at) = (0, ... , 1/(dt -1), ... , 0), and hence, A is isomorphie to the produet MV-algebra P. To prove uniqueness of the deeomposition (3.9), suppose

A = L b1

X ••• X

L bv

for suitable integers bi 2: 2. Then for each i = 1, ... , v, the kernel of the projeetion function (Xl, ... , x r ) ~ Xi is a maximal ideal I i of A. Moreover, AI I i ~ L bi and A ~ AlI1 x ... x AI Ir. 0 Lemma 3.6.6 Let A be a (nontrivial) MV-algebra, and J be an ideal 0/ A. Then the quotient algebra AI J is semisimple i/ and only i/ J is an intersection 0/ maximal ideals 0/ A.

Proof: Let h J : A - 4 AI J be the natural projeetion. Suppose that AI J is a semisimple MV-algebra. If the family of an maximal ideals of AI J, then

J

= h"J1( {O}) = h"J1(n Mi) = iEI

{MdiEI

nh·:/(M

i ).

iEI

denotes

3.7. BIBLIOGRAPHICAL REMARKS

75

By Proposition 1.2.10, J is an intersection of maximal ideals of A. Conversely, assume J to be an intersection of maximal ideals of A. Then J is the intersection of all the maximal ideals of A containing J. If {Mihel denotes this family, then, by Proposition 1.2.10, {hJ(Mi)heI is the family of all maximal ideals of AI J and hJ(J) = Rad(AI J). 0 The following result strengthens Proposition 3.6.1: Theorem 3.6.7 An MV-algebra A with K, many generators is semisimple if and only if for some nonempty closed subset X ~ [0, 1]/t, A is isomorphie to the MV-algebra of restrietions to X of all functions in Free/t. Proof: One direction is an immediate eonsequenee of Proposition 3.6.1. The other direction follows at onee from Propositions 3.1.5 and 3.4.5 and Lemma 3.6.6. 0 Corollary 3.6.8 For any MV-algebra A the following eonditions are equivalent:

(i) A is semisimple; (ii) A is isomorphie to a separating MV-algebra 01[0, 1]-valued eontinuous funetions on some nonempty eompaet Hausdorff spaee, with pointwise operations. 0 The following theorem direetly follows from Theorem 3.4.9 and Lemma 3.6.6. Theorem 3.6.9 Suppose A A is semisimple. 0

3.7

~

Free/tl J, with J a prineipal ideal. Then

Bibliographical remarks

Our definitions of term and free algebras are particular eases of abstract definitions in Universal Algebra (see, for instance, [25], [104] or [150]). Lemma 3.1.9 is due to McNaughton [152]. The simplified proof presented here is due to Rose and Rosser [216]. The proof of the one

76

CHAPTER 3. FREE MV-ALGEBRAS

variable McNaughton theorem, based on Farey series and Schauder hats, is taken from [181]. Our FareYn is a variant, due to Stern and Brocot, of the traditional Farey sequence of order n + 1, considered in most textbooks in number theory. For a proof of the fundamental theorem on polyhedra see, for instance, [250] or [82, p.31]. The fact, mentioned at the beginning of Section 3, that every polyhedral complex can be refined to a simplicial complex without adding new vertices is also weIl known: see, for instance, [82, Theorem III, 2.6]. Proposition 3.3.1 can be found in McNaughton's paper [152]. Lemma 3.4.8 was first proved in [172, Proposition 2.4]. Theorem 3.4.9 is the algebraic counterpart of results of Hay, W6jcicki and Rose that will be presented in the next chapter. The proof given here is taken from [170]. The example after Theorem 3.4.9 is due to W6jcicki [243]. Theorem 3.5.1, Proposition 3.6.1 and CoroIlary 3.5.4 are due to Chang [36] (see also [13],[144]). Definition 3.6.3 and Proposition 3.6.4 are due to Rodriguez [212] (see also [132, Proposizione 7]). Proposition 3.6.5 appears in [144], [233], [212].

Chapter 4 Lukasiewicz oo-valued calculus Since every MV-term T is astring of symbols over a finite alphabet, one may naturally consider the following decision problem: does there exist an effective procedure (for definiteness, a Turing machine) deciding whether an arbitrary equation T = 1 holds in all MV-algebras ? More generally, given two terms (J and T, does there exist an effective procedure to decide whether the McNaughton function determined by (J belongs to the principal ideal determined by T in the free MV-algebra Free w ? These are respectively known as the word problem for free MValgebras, and the word problem for finitely presented MV-algebras. In this chapter we shall reformulate these problems in purely logical terms, within the infinite-valued sentential calculus of Lukasiewicz. We shall regard MV-terms as propositions, valid equations as tautologies, ideals as theories, word problems as decision problems in this calculus-in the traditional sense. Adopting this viewpoint, we shall need an equivalent reformulation of Chang's completeness theorem to the effect that all tautologies are obtainable from a certain set of initial tautologies (corresponding to the MV-axioms) by a finite number of applications of modus ponens. Free MV-algebras shall be re-obtained as algebras of propositions up to logical equivalence. Using the results of Chapter 3 we shall finally obtain an effective procedure to decide whether a proposition is a tautology, thus automatically giving a positive solution to the word problem for free MV-algebras. Moreover, the

77 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

78

word problem for finitely presented MV-algebras shall be reformulated in terms of logical consequence. We shall give a positive solution to the problem, after a detailed analysis of the subtleties of the notion of consequence in the infinite-valued calculus.

4.1

Many-valued propositional calculi

In the early twenties, Jan Lukasiewicz introduced systems of logic in which propositions admit as truth values real numbers between and 1. As main propositional connectives he considered implication - and negation -', as given by the following "truth tables", where x and y denote arbitrary elements of the real unit interval [0,1]:

°

(4.1)

x - Y =def min(1, 1- x

+ Y)

and

(4.2)

-,x =def 1 - x.

Along with this infinite-valued propositional system, Lukasiewicz also considered, for each natural number n ;::: 2, an n-valued system, in which the truth values are the rational numbers 1 n- 2 0, --1' ... ' - - ' 1,

n-

n-1

and the truth values for implication and negation are again given by formulas (4.1) and (4.2). Thus in particular, for n = 2 the only possible truth values are and 1, and the above formulas give back the truth tables for implication and negation in the classical propositional calculus, provided 1 is interpreted as true and as false. As the reader will recall, formula (4.2) gives the operation -, in the MV-algebra [0,1]. On the other hand, formula (4.1) can be written in terms of the MV-operations as follows:

°

(4.3)

x - Y = -,x EB Y

whence, (4.4)

xEBy

= -,x -

y.

°

4.1. MANY-VALUED PROPOSITIONAL CALCULI

79

Thus, the operations ..." Ei' and -+ on [0,1] have the same interdefinability properties as negation, disjunction and implication in classical logic. It is natural to consider Ei' as a disjunction connective. Accordingly, equation (1.2) suggests to consider <:) as conjunction. Given an arbitrary MV-algebra A, let us define the binary operation -+ on A by formula (4.3). That is, for all x,y E A, x -+ Y =def ...,xEi'y. Then, triviallY, equation (4.4) holds in A. In the next section· we shall consider an equivalent reformulation of MV-algebras in terms of the operations -+, ..., and the constant 1. This will more closely correspond to Lukasiewicz's original presentation of the infinite-valued calculus, which we shall now introduce formally as follows: Definition 4.1.1 As in the classical case, one starts from the finite alphabet

(Propositional) formulas are built, exactly as boolean propositions, from a denumerable set of propositional variables, X, XI, XII, ... , for short, Xo, Xl, X 2 , ••• , by means of the connectives of negation"" and of implication -+. In more detail, the set Form of formulas is given inductively as follows:

Fl) Each propositional variable X k is a formula. F2) If a is a formula, then ...,a is a formula. F3) Ha and ß are formulas, then (a

-+

ß) is a formula.

Notation: For each formula a, Var(a) will denote the set of all propositional variables occurring in a. Definition 4.1.2 Let A be an MV-algebra. Then an A-valuation is a function 11: Form -+ A satisfying the following properties, where a and ß denote arbitrary formulas:

VI)

1I(...,a) =del ""1I(a),

80

V2)

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

1/(0

-+

ß) =def 1/(0)

-+

1/(ß)·

Since unique readability also holds for formulas, any A-valuation 1/ is uniquely determined by its values 1/(Xo) , 1/(X1 ), .... An A-valuation 1/ is said to A-satisfy a formula 0 iff lI(a) = 1; 0 is an A-tautology iff 0 is A-satisfied by an A-valuations. Formulas a and ß are semantically A-equivalent iff 1/(0) = 1/(ß) for an A-valuations 1/. From condition (V2) we get that 0 and ß are A-equivalent iff both formulas 0 -+ ß and ß -+ 0 are A-tautologies. Let e be a set of formulas. We say that a formula 0 is a semantic A-consequence of e iff each A-valuation 1/ that A-satisfies an the formulas in e also Asatisfies o. Thus in particular, 0 is an A-tautology iff a is a semantic A-consequence of the empty set. Using the identities (4.3), (4.4), every formula 0 containing the variables Xl, ... , X k can be straightforwardly transformed into an MVterm Ta in the same variables. Conversely, upon replacing every occurrence of the constant 0 in a term T by, say, the formula -,(X ---+ X), then T is transformed into a propositional formula 0'T' We shall tacitly identify propositional formulas and MV-terms, using the maps 0 I - t Ta and T f--+ 0'T' Also, using all abbreviations introduced in Section 1.4, we shall freely write, e.g., 1 instead of -,-,(X ---+ X). As a first application of this identification, an easy induction on the number of connectives in 0 yields Proposition 4.1.3 (i) Let A be an MV-algebra and 0 be a formula, with Var(o) ~ {Xill ... , X ik }. Then for each A-valuation 1/ we have

(4.6)

1/(0) = oA(1/(Xi}), ... , 1/(Xik )),

where OA: A k

---+

(ii) A formula holds in A;

0

A is the term junction defined in Section 1.4; is an A-tautology if and only if the MV-equation

0 =

1

and ß are semantically A-equivalent iff the equation o = ß holds in A iff OA = ßA. 0

(iii) Formulas

0

For each integer n ~ 2, the n-valued Lukasiewicz propositional calculus deals with propositional formulas equipped with the relation of

4.1. MANY- VALUED PROPOSITIONAL CALCULI

81

semantic Ln-equivalence. In the Lukasiewicz infinite-valued propositional calculus one considers propositional formulas equipped with the relation of semantic [O,I]-equivalence. By Proposition 4.1.3, the Completeness Theorem (Theorem 2.5.3) has the following equivalent formulation: Theorem 4.1.4 A formula a is a [0, 1]-tautology ij, and only ij, for every MV-algebra A, a is an A-tautology. Thus, for any two formulas a and ß, we have a[O,I) = ß[O,I) iff a A = ßA for all MV-algebras A.

o

Since OUf main concern here is the infinite-valued calculus, by a valuation we shall henceforth mean a [0, 1]-valuation; by a tautology we shall understand a [O,I]-tautology; also, semantic [O,I]-equivalence and [O,I]-consequence, shall be simply referred to as semantic equivalence and semantic consequence, respectively. Notation: For each 8 ~ Form, 81= will denote the set of semantic consequences of 8. In particular, 01= will denote the set of all tautologies.

The last two results above allow us to identify term functions a[O,I) and semantic equivalence classes of propositional formulas. Recalling now our analysis of free MV-algebras in Section 3.1, one may reasonably expect that McNaughton functions play in the infinite-valued calculus the same role played by boolean functions in the classical propositional calculus. As an instance of this role, consider the proof of the following result, which will be considerably strengthened in Corollary 4.5.3 and in Theorem 4.6.10 below: Proposition 4.1.5 There exists an effective procedure (for definiteness, a Turing machine) enumerating all formulas that are not tautologies. Proof: Fixing an arbitrary lexicographic ordering of all strings over alphabet E, let 'l/JI, 'l/J2, ... be a list of all n-variable formulas 'l/Jt = 'l/Jt(X I , ... , X n ) E Form. Let similarly xl, x 2 , ... list all n-tuples x = Xl, ... , X n of rational numbers in the unit real interval [0,1]. Thus, Xi = (xL . .. , x~) for suitable X} E [0,1] n Q. For each i = 1,2, ... let vi be the valuation such that vi(Xj ) = X} for all j = 1, ... , n., and

82

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

vi(Xm ) = 0 for all remaining variables. Whenever a pair (i, 'ljJt) is such that vi('ljJt) < 1, put 'ljJt in the list of nontautologies. By Proposition

4.1.3 together with Theorem 4.1.4 and our analysis of free MV-algebras in Section 3, the desired conclusion now follows from the continuity of McNaughton functions: indeed, if a formula 'ljJ is not a tautology then there is a valuation v whose values are all rational, such that v( 'ljJ) < 1.

o

In the next sections we shall obtain an effective procedure to enumerate all tautologies.

4.2

Wajsberg algebras

Stone's representation theorem for boolean algebras allows one to visualize the connectives of negation, disjunction and conjunction of the classical propositional calculus as the set-theoretical operations of complement, union and intersection. In the infinite-valued case, since EB and 0, together with negation -, can express V and A, the (additive) operations EB and 0 are regarded as more fundamental than the (lattice ) operations V and A. Historically, Chang introduced MV-algebras as generalized boolean algebras; indeed, Chang's completeness theorem is a generalization of the fact that if an equation holds in the boolean algebra {O, 1}, then the equation holds in every boolean algebra. Already the proof of the onevariable case of McNaughton's theorem given in Section 3 enhances the role of the associative-commutative operation EB, giving normal form reductions of all elements of Freel as sums of Schauder hat functions. In a later chapter we shall establish a categorical equivalence between MV-algebras and abelian f-groups with strong unit (the latter being deeply related to the time-honoured theory of magnitudes). Altogether, the operations 0 and EB have a distinctly arithmetic, rat her than a settheoretic significance. On the other hand, the identities x ~ y = -,x EB y and x EB y = ,x ~ y show that there is no essential difference of expressive power between the additive connectives and the implication connective. It turns out that for our analysis of proofs and consequence in this chapter, it is more convenient to replace the 0 and EB operations by

4.2. WAJSBERG ALGEBRAS

83

the implieation eonneetive -. In this way one obtains the following equivalent variant of MV-algebras: Definition 4.2.1 A Wajsberg algebra (for short, a W-algebra) is a system A = (A, -",1), where A is a nonempty set, and the binary operation -, the unary operation ' and the distinguished element 1 satisfy the following equations: W1) 1-x=x W2) (x - y) - ((y - z) - (x - z)) W3) (x - y) - y

=

=

1

(y - x) - x

W4) (,x - ,y) - (y - x)

=

1.

Lemma 4.2.2 Let A be an MV-algebra, and put x and 1 =def ,0. Then (A, -, ,,1) is a W-algebra. 0

y =def ,x EB y

Our next task is to prove that eonversely, eaeh W-algebra beeomes an MV-algebra, onee equipped with the operations -', x 8 Y =dej -'x y and 0 =def ,1. Lemma 4.2.3 Let A = (A, -,1) be a system satisfying (Wl), (W2) and (W3). Then the following properties hold fOT every x, y and z in A: W5) x - x

=

W6) Ifx - y

W7) x -1

1

= y - x = 1, then x = y

=1

W8) x - (y - x) W9) If x - y W10) (x

---4

y)

=

=Y

---4

---4

((z

W11) x - (y - z)

1 Z

= 1, then x - z

---4

x)

---4

(z - y))

= y - (x - z).

= =

1 1

CHAPTER 4. LUKASIEWICZ 00- VALUED CALCULUS

84

Proof: W5) By (W2), (1 -+ 1) -+ ((1 by (Wl), x -+ x = 1 -+ (x -+ x) = 1.

-+

x)

-+

(1

-+

x)) = 1, and then,

W6) If x -+ y = y -+ x = 1, then by (Wl) and (W3) we have x = 1 -+ x = (y -+ x) -+ x = (x -+ y) -+ y = 1 -+ Y = y. W7) By (W3), (Wl) and (W5), (x -+ 1) -+ 1 = (1 -+ x) -+ x = x x = 1. From this identity, (W2), (Wl) and (W5) we have 1 = (1 x) -+ ((x -+ 1) -+ (1 -+ 1)) = x -+ ((x -+ 1) -+ 1) = x -+ 1. W8) By (W2), (W7) and (WI), 1 = (y -+ 1) x)) = 1 -+ (x -+ (y -+ x)) = x -+ (y -+ x).

-+

((1

-+

x) -

-+ -+

(y -

W9) If x -+ y = y -+ z = 1, then by (W2), 1 = (x -+ y) -+ ((y - z) -+ (x -+ z)) = 1 -+ (1 -+ (x -+ z)), and applying (Wl) twice, we obtain x-+z=1. WlO) As a preliminary step, we shall prove the following weaker form of (Wll): Wll') If x

-+

(y

-+

z)

= 1, then y -

(x

-+

z)

= 1.

Suppose that x -+ (y - z) = 1. Replacing y by y -+ z in (W2) we get 1 = (x -+ (y -+ z)) -+ (((y -+ z) - z) -+ (x -+ z)) = 1 - (((y z) - z) -+ (x -+ z)). Hence, by (Wl) and (W3), ((z -+ y) -+ y) -+ (x - z) = 1. On the other hand, by (W8), y -+ ((z - y) - y) = 1. Applying (W9) to these identities, with x = y, y = (z - y) - y and z = x -+ z, we obtain y -+ (x -+ z) = 1. Now (WIO) follows at onee from (Wll') and (W2).

Wll) By (W3) and (W8), y -+ ((y -+ z) -+ z) = y -+ ((z -+ y) - y) = 1, and by WlO), ((y -+ z) -+ z) -+ ((x -+ (y -+ z)) -+ (x - z)) = 1. Applying (W9) to the last two identities, we obtain y -+ ((x - (y -+ z)) -+ (x -+ z)) = 1. Hence, by (Wll'), (x -+ (y - z)) -+ (y -+ (x - z)) = 1. By interchanging x and y in this identity we also obtain (y -+ (x -+ z)) -+ (x -+ (y -+ z)) = 1, and then we can apply (W6) to obtain (Wll). 0 Lemma 4.2.4 The following equations hold in every W-algebra A:

(i) -d

-+

x = 1;

4.2. WAJSBERG ALGEBRAS

85

(ii) -,x = x - -,1;

(m) -,-,x = x; (iv) x - y = -'y - -,x. Proof: (i) By (WlO) we have ((-,x - -,1) - x) - ((-,1 - (-,x -,1) - (-,1 - x» = 1. Since, by (W1) and (W4), (-,x - -,1) - x = (-,x - -,1) - (1 - x) = 1 and, by (W8), -,1 - (-,x - -,1) = 1, whence the desired conclusion follows from (W1). (-,-,1 - -,x) = 1. By (W4), (-,-,1 - -,x) (x - -,1) = 1. Then from (W9) we obtain -,x - (x - -,1) = 1, and applying (W11) we have x - (-,x - -,1) = 1. As seen in the proof of (i), we also have (-,x - -,1) - x = 1. Hence by (W6) we obtain x = -,x - -,1, and taking into account (W3), together with (i) and (W1) we get x - -,1 = (-,x - -,1) - -,1 = (-,1 - -,x) - -,x = 1 -

(ii) By (W8), -,x -

(iii) By (ii), (W3), (i) and (W1) we can write -,-,x -,1 = (-,1 - x) - x = 1 - x = x.

= (x - -,1) -

(iv) By (W4) and (iii) we have 1 = (-,-,x - -,-,y) - (-,y - -,x) (x - y) - (-,y - -,x). Nowapply (W4) and (W6). 0

=

Theorem 4.2.5 Let (A, -, -,,1) be a Wajsberg algebra. Upon defining x EB y =dej -,x - y and 0 =dej -,1, the system (A, EB, -', 0) is an MValgebra. Proof' (A, EB, -', 0) satisfies MVl), because x EB Y

= -,x -

y

=

-,y - -,-,x

= Y EB x,

and also satisfies MV2), as shown by the following identities: x EB (y EB z)

= -,z -

= -,x - (-,y - z) = -,x - (-,z - y)

(-,x - y)

= z EB (x EB y) = (x EB y) EB z.

The remaining verifications, MV3)-MV6), are easy consequences of the properties established in the previous lemmas. 0

86

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

In what follows, ---.. will denote the binary operation defined on MV-algebras by formula (4.3). In the light of the above theorem, this operation satisfies properties (W1) - (W11). Lemma 1.1.4(iii) gives the following remarkable relation between the operations ---.. and 0 on an MV-algebra A: (4.7)

For all x, y, Z E A, z 0 x::; y iff z::; x ---.. y.

Definition 4.2.6 An implicative filter of an MV-algebra Ais a subset F of A satisfying the following conditions:

(Fl) 1 E F; (F2) For all x, y in A, if x

E Fand

x ---.. y E F, then y E F.

An implicative filter F of A is said to be proper iff F =I- A; F is called maximal iff F is proper and A is the only implicative filter strictly containing F.

Lemma 4.2.7 The following are equivalent conditions for each subset F of an MV-algebra A:

(i) F is an implicative filter;

(ii) F =I 0; if x

E Fand

x ::; y E Athen y E F; if x, Y E F then

x0y E F;

(iii) The set -,F =def {-,x I x E F} is an ideal of A. Proof: (i) implies (ii): Let F be an implicative filter. By (F1), 1 E F. Suppose x E Fand x ::; y. Since this last condition is equivalent to x ---.. Y = 1, (F2) yields y E F. Note that y ---.. (x ---.. (x 0 y» = 1. Therefore if x, y are both in F, then by (F2) we obtain that x 0 y E F. (ii) implies (iii): This immediately follows by definition of the operation 0 as given by equation (1.2). (iii) implies (i): Suppose that -,F is an ideal. Since 0 E -,F, we have 1 E F. Suppose now that x and x ---.. y are in F. This means that -,x E -,F and xe y E -,F, and then -,x V -,y = ...,x EB (x e y) E ...,F. Therefore, -,y E -,F, i.e., y E F. 0

4.3. PROVABILITY

87

The above lemma allows us to obtain properties of implicative filters from properties of ideals. For instance, if h: A --+ A' is a homomorphism then the set F(h) =del {x E AI h(x) = I} is an implicative filter; moreover, h(x) = h(y) Hf (x --+ y) 1\ (y --+ x) E F(h).

Notation: Given an implicative filter F, by abuse of notation we shall write AI F to denote the quotient algebra AI...,F. The intersection of any nonempty family of implicative filters of an MV-algebra A is an implicative filter, and since A itself is an implicative filter, we can give the following Definition 4.2.8 For any subset X of an MV-algebra A, the implicative filter V(X) generated by X is the intersection of all implicative filters containing X. Proposition 4.2.9 Let A be an arbitrary MV-algebra. (i) Let s, t E A and Y ~ A. Then t E V(Y U {s}) iff there is an integer n;::: 1 such that sn --+ t E V(Y); (ii) For any s, t E A we have t E V( {s}) iff there is an integer n ;::: 1 such that sn --+ t = 1; (iii) For arbitrary Xl, ... , Xk E A we have

(4.8) V( {Xl,"" xd) = V(XI 0 ... 0 Xk).

Praof (i) and (ii) can be easily derived from (1.8) and (1.9), respectively. To prove (iii), let nl,"" nk ;::: 1 be integers, and n = max(nl," ., nk). Then the desired conclusion follows from the inequality (Xl 0 ... 0 Xk)n ~ X~l 0 ... 0 X~k. 0

4.3

Provability

Definition 4.3.1 An axiom of the Lukasiewicz infinite-valued propositional calculus is a formula that can be written in any one of the following ways, where a, ß and I denote arbitrary formulas:

(Al) a

--+

(ß

--+

a);

CHAPTER 4. LUKASIEWICZ 00- VALUED CALCULUS

88

(A2) (a

~

(A3) «a

(A4) (-,a

ß)

~ ~

~

ß)

~

-,ß)

(a

«ß

~

,)

ß)

~

«ß ~ a)

(ß

~

~

~

~

,));

~

a);

a).

Note that one can effectively decide whether a given string of symbols over the alphabet E of (4.5) is an axiom.

Definition 4.3.2 A prooffrom a set e of formulas is a finite string of formulas al," . ,an, with n ~ 1, such that, for each 1 :::; i :::; n:

(i) (ii)

ai is an axiom, or ai E

e, or

(iii) there are j, k E {I, ... , i-I} such that formula (aj

~

ak coincides with the

ai).

A formula a is provable /rom e, in symbols, e f- a, iff there is a proof e, such that an = a. Equivalently, we say that a is a syntactic consequence of e. The set of provable formulas from e shall be denoted al,' .. ,an from

As usual, we say that formula ß follows by modus ponens from formulas a and a ~ ß. With this terminology, condition (iii) in the above definition can be given by the following equivalent reformulation: (iii / ) there are j, k E {I, ... , i-I} such that ai follows by modus

ponens from

aj

and ak.

In Definition 4.3.2 the case condition (ii) becomes vacuous.

e

= 0 is

not excluded. In this case,

Definition 4.3.3 By a proof we shall henceforth me an a proof from the empty set; by a provable formula we shall mean a formula that is provable from the empty set. We shall also write f- a in place of 0 f- a.

89

4.3. PROVABILITY

Given a finite sequence of strings of symbols from the alphabet E we can effectively decide whether such string is a proof. Using some lexicographic ordering, we then see that there exists an effective procedure to list the set of provable formulas. Our next aim is to show that the tautologies coincide with provable formulas; this, together with our discussion in the first sections, will give a positive solution to the decision problem for the infinite-valued calculus of Lukasiewicz. Notation: We shall use CiVß and CiAß as abbreviations of (Ci --+ ß) --+ ß and ""((""Ci --+ ...,ß) --+ ...,ß), respectively. With this notation, (A3) can be written as (Ci V ß) --+ (ß V Ci). Proposition 4.3.4 For all formulas Ci, ß and, we have

(4.9)

f- (Ci --+ (ß --+ ,)) --+ (ß --+ (Ci --+ ,))

(4.10) f- Ci

Ci

--+

(4.11) f- (ß

--+ ,) --+

(4.12) f- ...,...,a

--+

(ß

--+

...,ß)

(4.15) f- (a

--+

...,...,ß)

--+

--+

--+

ß)

--+

(Ci

--+ ,))

a)

---t

(4.14) f- (a

(4.16) f- a

((Ci

(ß

--+

--+

(...,ß

""Ci) --+

""Ci)

...,...,a.

Proof: To prove (4.9) it suffices to check that the following string of formulas is a proof. To help the reader, on the right-hand side of each formula occurring in the string, we shall point out whether this formula is an axiom, or else the formula follows by modus ponens from preceding formulas. al : ß --+

Ci2 : (ß

h V ß) = ß --+ (h --+ ß) --+ ß)

--+ (,

V ß))

--+ (((,

V ß)

--+

(ß V,))

(Al) --+

(ß

--+

(ß V,)))

(A2)

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

90

a3 : ((, V ß) a4 :

b

as : ß

V

ß)

-+

a6 : (ß

-+

(ß V,))

-+

(MP-al, a2) (A3)

(MP- a4, a3)

(ß V ,)

-+

(ß V,))

V ,) -+

a8 : (a

(ß

-+

= (a

-+

-+

(a

(((ß V ,)

-+

(a

-+ ,)) -+

(ß

-+

(a

-+ ,))

-+ ,)) -+

(((ß

-+ ,)) -+ -+

(ß

-+

((a

(a7

alO : a7

-+

((a

(a

-+

(ß

an :

(ß

-+

(ß V ,)

-+

a7 : ((ß

ag : a8

(ß V,))

-+

-+ ,)) -+

(ß

-+

(ß

((ß V ,)

(ß

(a

-+

(ß

(a

-+ ,)))

-+

(a

-+ ,)))

(a

(A2)

-+ ,))))

(MP-a8, ag) (MP- a7, alO)

-+ ,))

This settles (4.9). To prove (4.10) note first that, taking , (4.9), we obtain I- (a

If al,"" an

= (a

-+

a))

-+

(ß

-+

(ß

-+

a))

-+

(A2)

(A2)

-+ ,))

(ß

-+

(a

-+ ,))

-+ ,)) -+

-+ ,)) -+

-+ ,)) -+

-+

-+

(MP-as, a6)

(a

-+ ,) -+ ,) -+

(ß

= a

-+

(ß

-+

(a

-+

(ß

-+

(a

-+

a)) is a proof, then

in

a)).

is also a proof. Hence taking any axiom as ß, we obtain that the sequence al,' .. , an> an+l, an+2, a n+3 = ß, a n+4 = a -+ a is a proof of a -+ a. To prove (4.11), as a particular case of (4.9) we have I- {) = ((a

((ß If al, .... an

-+

ß)

-+

((ß

-+ ,) -+

-+ ,) -+ (( a -+

ß)

-+

(a

(a

= {) is a proof, then the string

-+ ,))) -+

-+ ,))).

4.3. PROVABILITY O:n+2 = (ß

--+

I)

91 --+

((0:

--+

ß)

--+

(0:

I))

--+

is also a proof. This settles (4.11). To prove (4.12) it is enough to check that the following string of formulas is a proof:

(Al)

0:2 : (··ß

--+ • • 0:) --+

0:3 : ("0:

--+ ( • •ß --+ • • 0:)) --+ (( ( • •ß --+ • • 0:) --+

(,0: 0:4 : ((··ß

(.0:

.ß))

--+

--+

--+

·ß)

(.-;'0:

--+

(.0: --+ ·ß)) (.0: --+ .ß))

--+ • • 0:) --+

(••0:

(A4)

--+

(.0:

--+

·ß)))

(A2)

--+

(MP-0: 1,0:3) (MP- 0:2,0:4)

0:6 : (,0:

--+

0:7: ("0:

·ß)

--+

--+

(ß --+ 0:)

(A4)

(.0: --+ ·ß)) --+ (((,0: (••0: --+ (ß --+ 0:)))

0:8 : ((,0:

--+

·ß)

0:9 : ••0:

--+

(ß

--+

--+

(ß

0:))

--+

--+

--+

·ß)

--+

(ß

--+

0:))

--+

(A2)

("0:

--+

(ß

--+

(MP- 0:5,0:7)

0:))

0:)

(MP- 0:6,0:8).

This settles (4.12). To prove (4.13), combining (4.9) and (4.12) we easily obtain f- ß("0: --+ 0:). Since ß is arbitrary, letting ß be an axiom we immediately get the desired proof. To prove (4.14) it is enough to verify that a new proof is obtained upon adding to any proof 0:1, ... , O:n of ••0: --+ 0: the following formulas:

O:n+1: (••0:

--+

O:n+2 : (0:

·ß)

--+

O:n+3: ((0:

--+

0:)

--+

--+

·ß)

--+

((0:

--+

.ß)

--+ ( • •0: --+

·ß))

(A2)

("0: - ·ß) (,,0:

--+

·ß))

--+ ((( • • 0: --+

·ß)

--+

92

CHAPTER 4. LUKASIEWICZ 00- VALUED CALCULUS

(ß - -'0)) - ((0 - -.ß) - (ß - -'0)))

(A2)

on+4 : ((-'-'0 - -.ß) - (ß - -'0)) ((0 - -.ß) - (ß - -'0))

(A4) a n +6 : (0 - -.ß) - (ß - -.a) (MP- 0n+5, On+4). To obtain (4.15) it is sufficient to replace ß by -.ß in (4.14). Finally, to prove (4.16), replacing 0 in (4.14) by -.a and ß by 0, we get f- (-.a - -'0) - (a - -'-'0). Combining now this result with (4.10), in which 0 is replaced by -'0, we get the desired conclusion. 0 From (Al) and modus ponens we immediately obtain then, for each ß,

ß - o.

(4.17)

If f-

4.4

Lindenbaum algebra

0

f-

Theorem 4.4.1 Let the binary relation = on Form be defined by o = ß iff f- 0 - ß and f- ß - o. Then = is an equivalence relation, called syntactic equivalence , satisfying the following conditions: (4.18) If a (4.19) If 0

= 'Y and ß = 0 then (0 -

=ß then =-.ß.

ß)

= (-y -

0);

-'0

Proof" It is obvious that 0 = ß implies ß = 0 and, by (4.10), 0 = o. By (A2), {o - ß,ß - 'Y} f- 0 - 'Y and {'Y - ß,ß - o} f- 'Y - o. Hence, = is transitive, and we have shown that = is an equivalence relation on Form. By (A2), for any formulas 0, ß and 'Y we have {'Y - o} f- (a - ß) - ('Y - ß)

and

{o - 'Y} f- ('Y - ß) - (0 - ß). Therefore, (4.20) If a

='Y then

0 -

ß

='Y -

ß·

4.4. LINDENBAUM ALGEBRA

93

Analogously, from (4.11) we obtain (4.21) If ß

=0 then 'Y

-+

ß =: 'Y -+ o.

Now (4.18) follows from (4.20), (4.21) and the transitivity of ==. By (4.13) and (4.16), for every formula ß, ß == "ß. Hence by (4.21), for arbitrary formulas a and ß, a -+ ß == a -+ "ß. Therefore, r (a -+ ß) -+ (a -+ "ß). From this result and (4.15) we obtain r (a -+ ß) -+ (,ß -+ ,a). Since by (A4), r (,ß -+ ,a) -+ (a -+ ß), we have (4.19). 0 Notation: The equivalence dass of formula a with respect to syntactic equivalence shall be denoted by lai. In symbols,

lai =def {ß E Form I ß Lemma 4.4.2 For each formula a, a E 0f- iJJ

a}.

lai = 0f-.

Prool If lai = 0f- then a E 0f- because a E lai. Conversely, suppose a E 0f-, Le., r a. Take ß E 0f-. By (4.17), ß == a. Hence 0f- ~ lai. Now take ß E lai. Then r (a -+ ß) and, by modus ponens, r ß. Hence

lai ~ 0f-.

0

Theorem 4.4.3 The quotient set Form/=: becomes a Wajsberg algebra, once equipped with the operations -+ and ' and the constant 1 as given by the following stipulations: (4.22)

lai -+ IßI =def la -+ ßI

(4.23)

,lai =def I,al

(4.24) 1 =def 0f-. Prool By (4.18) and (4.19) in Theorem 4.4.1, it follows that the two identities (4.23) and (4.24) yield weIl defined operations on the quotient set Form/ =:. Moreover, by Lemma 4.4.2, 0f- E Form/ ==. Then it remains to verify that the operations defined by (4.22) and (4.23) and the constant 1 as defined by (4.24) satisfy equations (W1) - (W4) in Definition 4.2.1. To prove (W1), note first that 1 -+ lai = 0f- -+ lai =

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

94

Iß

al,

where ß E 0f-. By (Al) we have f- a --+ (ß --+ a). On the other hand, by (A3), (ß --+ a) --+ a (a --+ ß) --+ ß and, since f- ß, it follows from (4.17) that f- (a --+ ß) --+ ß. Hence, by Lemma 4.4.2 we have f- (ß --+ a) --+ a, and then ß --+ a = a. Consequently, 1 --+ lai = 0f- --+ lai = Iß --+ al = lai, and this proves (W1). In the light of Lemma 4.4.2, conditions (W2), (W3) and (W4) followat once from (A2), (A3) and (A4), respectively. 0 --+

=

By Theorem 4.2.5 we get

Corollary 4.4.4 The quotient set Form/ = becomes an MV-algebra with the operations. and Ee and the constant 0 defined by (4.25)

·Ial =def I·al

(4.26)

lai EeIßI =def I·a --+ ßI

(4.27) 0 =def

.0f- = {a

E

Form I there is ß E 0f- such that a

=·ß}·

o The MV-algebra

.c =def (Form/=, 0", Ee)

is called the Lindenbaum algebra sitional calculus.

4.5

0/ Lukasiewicz infinite-valued propo-

All tautologies are provable

Lemma 4.5.1 Each /ormula provable from a set a semantic consequence 0/ this set. In symbols,

e 0/ /ormulas is also

Proof: Let v: Form --+ [0,1] be a valuation such that v(a)

=

1 for all a E e. By induction on n we shall prove that if al," . ,an is a proof from e, then v( an) = 1. If n = 1, then al is either an axiom or it belongs to e. In the second case, v(ad = 1 by the hypothesis on v. In the first case we also

4.5. ALL TAUTOLOGIES ARE PROVABLE

95

have lI(al) = 1 because all axioms are tautologies, as shown by direct inspection of (Al) - (A4). Let n > 1 and suppose that, for each proof from 8, ßl,"" ßm, with m < n, we have lI(ßm) = 1 (induction hypothesis). Let al,' .. , an be a proof from 8. If an is not an axiom and does not belong to 8, then there are i, j E {I, ... , n} such that aj coincides with the formula (ai - an). Since both al,' .. ,ai and al, ... , aj are proofs from 8, by induction hypothesis, lI(ai) = lI(aj) = 1. Therefore, 1 = lI(aj)

= 1 - lI(an) = lI(a n).

o Thus, in particular, all provable formulas are tautologies. The converse follows from Chang's completeness theorem:

Theorem 4.5.2 Every tautology is provable. Thus, tautologies eoineide with provable formulas, in symbols, 0F = 0'r-. Proof" First of all , for each propositional variable Xi, the syntactic equivalence class lXii is an element of the Lindenbaum algebra C. Let a be a formula with Var( a) ~ {Xit , ... , X in }. By an easy induction on the number of connectives in a, it follows that aC(IXi11, ... , IXin I) = lai. Thus, in case a E Form is not provable, then by Lemma 4.4.2 and (4.24), lai =11, whence aC(IXitl,.··, IXinl) =11. Stated otherwise, the Lindenbaum algebra C does not satisfy the equation a = 1. Hence by Chang's completeness theorem 2.5.3, the MV-algebra [0,1] does not satisfy this equation, Le., a is not a tautology. The rest follows from Lemma 4.5.1. 0

Corollary 4.5.3 There is an effeetive proeedure (for definiteness, a Turing maehine) deeiding whether an arbitrary formula is a tautology in the infinite-valued ealeulus of Lukasiewiez. Proof: By the above theorem, together with the remark following Definition 4.3.3, tautologies can be effectively enumerated. The effective enumerability of nontautologies was shown in Proposition 4.1.5. 0

This result shall be considerably strengthened in a later chapter, where we shall give a more accurate estimate of the number of Thring steps needed to decide whether a formula is a tautology.

96

CHAPTER 4. LUKASIEWICZ 00- VALUED CALCULUS

Corollary 4.5.4 The relations of semantic and syntactic equivalence coincide: thus, for arbitrary formulas a and ß, a[O,lj = ß[O,lj iff

a=ß·O Following tradition, whenever two formulas are syntaetieally (= semantically) equivalent, we shall heneeforth say that they are logically equivalent.

Proposition 4.5.5 Up to isomorphism, the Lindenbaum algebra L coincides with the free MV-algebra over the generating set {IXoI, lXII, ... } of logical equivalence classes of propositional variables. Proof: Reealling the terminology introdueed in Seetion 3.1, let us define the map c.p: L - t Term([O, 1], w) by the stipulation c.p(lal) = a[O,lj. Then Corollary 4.5.4 implies that c.p is an isomorphism of the Lindenbaum algebra L onto the term algebra Term([O, l],w). In particular, the rest riet ion of c.p to the set of logical equivalenee classes of propositional variables yields a bijeetion from this set onto the set of projeetion functions {-lro, 'lr1, ... }. Henee, the desired eonclusion follows from Proposition 3.1.4. 0

Lemma 4.5.6 Let A be an MV-algebra and v: Form valuation. Then the stipulation

-t

A be an A-

defines a homomorphism h v : L - t A. Conversely, for each homomorphism h: L - t A, the stipulation vh(a) = h(lal) defines an A-valuation VA: Form - t A. Moreover, the correspondence v ~ h ll is a one-one mapping from the set of A-valuations onto the set of homomorphisms into A of the Lindenbaum algebra L. The inverse mapping is given by h ~ Vh. Proof: Sinee ß E lai implies v(ß) = v(a), the stipulation hll(lal) = v(a) defines a mapping h v : L - t A. The rest of the proof is obvious.

o

4.6. SYNTACTIC AND SEMANTIC CONSEQUENCE

4.6

97

Syntactic and semantic consequence

By Lemma 4.5.1, for any set e of formulas we have the inclusion er- ~ By Theorem 4.5.2, 0F = 0r-. As we shall see in this seetion, in general, eF =I- er-; we shall give neeessary and suffieient eonditions for eF to eoincide with er-.

eF.

Definition 4.6.1 A theory of Lukasiewiez infinite-valued propositional ealeulus is a set e of formulas satisfying the following eonditions: (Tl) All axioms belong to (T2) If a E

e and (a -+

e;

ß) E

e, then

ß E

e.

Proposition 4.6.2 The following conditions hold for each set formulas:

(i)

er-

(ii)

e

is the smallest theory containing is a theory iff

e

of

e;

e = er-;

(iii) 1f e is a theory and a E

e,

then

lai

~

e.

Praof: (i) It follows at onee from Definition 4.6.1 that er- is a theory. Suppose that :E is a theory and e ~ :E. If al, ... , an is a prooffrom e, arguing as in the proof of Lemma 4.5.1, we see that an E :E. Therefore, er- ~ :E. The proof of (ii) trivially follows from (i). To prove (iii), note that if ß E lai then a -+ ß E er- and, by (T2), ß E e. 0

Let

{eihEI be a nonempty family of theories.

It is easy to see that is a theory; furt her , from the above proposition we immediately get that (UiEI is the smallest theory eontaining ei for all i E 1. Thus, the set of theories of Lukasiewicz infinite-valued calculus, ordered by inclusion, is a eomplete lattiee. We shall denote this lattice by Theo.

niEI ei

eit

Comparing Definitions 4.2.6 and 4.6.1 we immediately get Theorem 4.6.3 The correspondence (4.28)

e

1-+

lei =def {lai

E J:,

I a E e}

98

CHAPTER 4. LUKASIEWICZ

00- VALUED

CALCULUS

defines an isomorphism from the lattice Theo onto the lattice of implicative filters of the Lindenbaum algebra C. The inverse isomorphism is given by

(4.29) F

1-+

{a E Form Ilal E F}. 0

An important result of classical propositional calculus is the Deduction Theorem. The next proposition gives aversion of this theorem for the Lukasiewicz infinite-valued propositional calculus.

Proposition 4.6.4 For arbitrary formulas a and ß, and for each Form we have

e~

ßE(8U{a}t ij, and only ij, there is an integer n ~ 1 such that (an - ß) E 8f-. Prool Since ß E (8U{a}t iff IßI E 1(8U{a}tl, the result follows from Proposition 4.2.9 and 4.6.3. 0

Corollary 4.6.5 ß E {at iJJ there is an integer n > 1 such that an _ ß is a tautology. Prool By Proposition 4.2.9(ii). 0 The next theorem gives the desired necessary and sufficient condition for 8 F to coincide with 8f-.

Theorem 4.6.6 For each 8 ~ Form, 8 F = 8f- iJJ 18f-1 is an intersection of maximal implicative filters of the Lindenbaum algebra C. Proof: For each valuation v: Form - [0,1] we have

where h v : C - [0,1] is the homomorphism defined in Lemma 4.5.6. We next observe that whenever a does not belong to 8 F there exists a valuation Va: Form - [0,1] such that va(ß) = 1 for all ß E 8 and va(a) < 1. Therefore, 8F

=

n vc:- ({1}), 1

a~eF

4.6. SYNTACTIC AND SEMANTIC CONSEQUENCE

99

whence, by Theorem 4.6.3 and equation (4.30),

Consequently, if e~ = e~, then le~1 is an intersection of maximal implicative filters of .c. Conversely, suppose there is a family {MihEI of maximal implicative filters of .c such that le~1 = niEI Mi. Since for each i E I the quotient MV-algebra .clMi is simple, there is a homomorphism hi:.c- [0,1] such that Mi = hi 1 ({1}). By Lemma 4.5.6, for each i E I there is a valuation Vi: Form - [0,1] such that hi = h Vi • Hence, again using Theorem 4.6.3 and (4.30), we conclude that e~

=

n

iEI

Vi- 1 (

{1 }).

Thus, whenever a
Theorem 4.6.7 For each finite set

e ~ Form,

e~

= e~.

Definition 4.6.8 For each theory e, the MV-algebra the Lindenbaum algebra ofe, and is denoted by .c(e).

.c.

0

.cIlei

is called

100

CHAPTER 4. LUKASIEWICZ

00- VALUED

CALCULUS

Note that .c ~ .c/10'·l Therefore,.c is the Lindenbaum algebra of the theory 0r-. From Theorem 4.6.6, together with our criterion for semisimplicity, (Theorem 3.4.9), for any set of formulas 8 we also have: 8r- = 81= Hf .c( 8) is semisimple. Theorem 4.6.9 Up to isomorphism, every countable MV-algebra A is the Lindenbaum algebra 01 same theory in the infinite-valued calculus 01 Lukasiewicz built /rom the variables X o, Xl, .... Proo/: Let g: {IXol, lXII, ... } -+ A be an arbitrary map from the set of logical equivalence classes of propositional variables, onto A. By Proposition 4.5.5, 9 can be extended to a (unique) surjective homomorphism h:.c -+ A. Let F = {lai E.c I h(laJ) = I}. Then by Theorem 4.6.3, F corresponds to a theory 8, and from A ~ .cl F we conclude A = .c(8).

o

Remark: A straightforward generalization of the above theorem shows that every MV-algebra arises as the Lindenbaum algebra of so me theory, provided sufficiently many variables are available in the alphabet.

The logical counterpart of the ward problem lor finitely presented MV-algebras, mentioned in the Introduction, has the following uniformly positive solution, yielding a generalization of Corollary 4.5.3: Theorem 4.6.10 There is a Turing machine U such that, over any input (8, 'I/J), where 8 is a finite set ollormulas and'I/J is a lormula, U decides in a finite number 01 steps whether or not 'I/J is a syntactic (= semantic) consequence 018. Proo/: By (4.8) and Theorem 4.6.7 we can safely assurne that 8 only contains a single formula () = (}(XI , ... , X n ). In light of Proposition 4.2.9(ii), we can effectively enumerate all pairs ((), 'I/J) such that 'I/J is a consequence of (). Conversely, we shall now describe an effective enumeration of all pairs ((), 'I/J) such that formula 'I/J is not a consequence of (). Arguing as in the proof of Proposition 4.1.5, first of all we shall effectively enumerate all triplets (x, (),
(i) x is an m-tuple of rational numbers in [0,1] (m 2: n), and,

101

4.7. BIBLIOGRAPHICAL REMARKS

(ii) letting Je and JCP be the McNaughton functions corresponding to B and 'P via Propositions 3.1.8 and 4.5.5, we have fe(x) = 1 and JCP(x) < 1. By Theorem 4.6.7, the existence of such rational point x is a sufficient condition for 'P not to be a consequence of B. Claim. The existence of x E [0, l]m n Q is also a necessary condition for 'P not to be a consequence of B. As a matter or fact, assurne 'P is not a consequence of B, and let Y E [O,I]m be an m-tuple of real numbers such that Je(Y) = 1 and JCP(Y) < 1. It is sufficient to argue in case m = n. Let jcp be the restriction of Jcp to the set Z =def {z E [0, Ir

I

Je(z) = I}.

By Proposition 3.1.8 JCP is a continuous, piecewise linear function, each piece having integer coefficients. Arguing as in Proposition 3.3.1, we see that the minimum value of jcp is attained at some point Z E Z with rational coordinates. Our claim is settled. To complete the proof, from the above list of triplets (x, B, 'P) it is now easy to construct the required effective enumeration of all pairs (B, 'P) such that 'P is not a consequence of B. A routine argument now yields the desired Turing machine U deciding whether 'P is a consequence of B. 0

4.7

Bibliographical remarks

The paper [139] is a basic reference for Lukasiewicz infinite-valued calculus. The equivalence between MV-algebras and Wajsberg algebras was first proved by Rodriguez in [212] (see also Font, Rodrfguez and Torrens [86], and Komori [130]). The crucial distinction between syntactic and semantic consequence relations in the infinite-valued ca1culus is due to W6jcicki [243] and [244, Theorem 4.3.4]. The fact that the two notions coincide for finitely axiomatizable theories was first proved by Hay in [113]. A related result by Rose is in [215, Lemma 1]. A geometrie visualization of the two notions, using the differential structure of McNaughton functions was given in [170].

102

CHAPTER 4. LUKASIEWICZ oo-VALUED CALCULUS

For the proof of the completeness theorem, following Chang [38] we use examples of proofs already present in [216]. Historically, the original list ofaxioms conjectured by Lukasiewicz to be complete for the infinite-valued calculus also included the axiom

whose redundancy was proved simultaneously by Meredith [154] and Chang [37].

Chapter 5 Ulam's game The crucial problem of interpreting n truth values when n > 2 was investigated, among others, by Lukasiewicz hirnself. As shown in this chapter, a simple interpretation is given by Ulam game, the variant of the game of Twenty Questions where n - 2 lies, or errors, are allowed in the answers. The case n = 2 corresponds to the traditional game without lies. The game is originally described by Ulam on page 281 of his book [235] as follows: Someone thinks of a number between one and one million (which is just less than 220). Another person is allowed to ask up to twenty questions, to each of which the first person is supposed to answer only yes or no. Obviously the number can be guessed by asking first: Is the number in the first half million? then again reduce the reservoir of numbers in the next quest ion by one-half, and so on. Finally the number is obtained in less than log2(lOOOOOO). Now suppose one were allowed to lie once or twice, then how many quest ions would one need to get the right answer?

5.1

Questions and answers

Let us inspect a round of Ulam game: initially, the two players agree to fix a nonempty finite set S of numbers, called the search space, and an integer m 2: O. Then the first player chooses a number x E S, and the second player must find the unknown x, by asking the smallest possible number of questions, to each of which the first player can only answer 103 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

104

CHAPTER 5. ULAM'S GAME

"yes" or "no", being allowed at most m errors/lies in his answers. By definition, a question is a subset of S: thus for instance, the quest ion is x an even number ?

is identified with the set of all even numbers in S. We can safely assurne Pinocchio to be the first player, and identify ourselves with the second player. Pinocchio's answers are propositions of either form "yes, (x is even) " , or "no, (x is odd)". Dur state of knowledge about x is uniquely determined by the conjunction of these answers. The latter, in general, do not obey the rules of classicallogic, for (a) The conjunction 01 two equal answers to the same repeated question need not be equivalent to a single answer. Thus, the classical idempotence principle fails. To see this, let us assurne that Pinocchio can lie at most once. Suppose we ask twice the following quest ion "is x even?". If the answer is "yes" in both cases, then x must be even. However, after the first answer we are not certain that x is even. (b) The conjunction 01 two opposite answers to the same repeated question need not lead to contradiction. To see this, again using the above example, if Pinocchio answers "yes" to the first question, and "no" to the second, then we cannot conclude that S is empty. In the traditional error-free game, our knowledge on x is usually represented by a function # : S ~ {O, I} giving, for every z E S, the number a = #z of falsified answers, where all a ;::: 1 are collapsed to 1. Stated otherwise, for each z E S, #z = 0 Hf z does not falsify any answer; #z = 1 iff z falsifies at least one answer. In Ulam game with m lies, our knowledge is also given by a function Cl : S ~ {O, 1, ... , m, m + I}, counting the number a = Cl(Z) of answers falsified by z, where all a ;::: m + 1 are collapsed to m + 1: thus Cl(Z) = 0,1, ... , m or m + 1, according as z falsifies 0,1, ... , m or ;::: m + 1 answers, respectively.

5.2

Dynamics of states of knowledge

Let us assurne that, after receiving a certain number of answers from Pinocchio, our knowledge is represented by the function Cl : S ~

5.2. DYNAMICS OF STATES OF KNOWLEDGE

105

{O, ... , m + I}. Let D be an arbitrary subset of S, and D = S\D its complementary subset. Suppose our quest ion "does x belong to D?", now receives from Pinocchio the answer "yes". Then how should we represent our new knowledge 0" ? Let the (m+2)-tuple (So, ... , Sm+l), be defined by

Si (i

= 0, ... , m +

=def {Z

E S I o'(z)

= i},

1). There is an obvious correspondence between

0'

and

(So, . .. , Sm+l), generalizing the correspondence between subsets and characteristic functions. The knowledge 0" = (Sb, ... ,S:n+l)' obtained

from by

(5.1)

0'

as a consequence of the positive answer to question D is given

Sb

=

So n D,

and, for each i

(5.2)

= 1, ... , m,

S; = (Si n D) U (Si-l n D).

As a matter of fact, whenever y E S is a valid candidate for the unknown x, the number o'(Y) of answers falsified by y will remain constant iff y satisfies D iff y E D; on the other hand, o'(Y) must be increased by one iff y falsifies D iff y rt. D. One can similarly deal with the negative answer to question D; such answer has precisely the same effect as a positive answer to the opposite quest ion D. Formulas (5.1) and (5.2) describing the dynamics of Ulam game acquire a particularly simple form if, instead of assigning to each y E S the number o'(Y) of falsified answers, we assign the truth value r(y) given by

(5.3)

r(y)

=def

1 - O'(y)/(m + 1).

Intuitively, the truth value r(y) measures, in units of m + 1, how distant y is from the condition of falsifying too many answers. More

106

CHAPTER 5. ULAM'S GAME

precisely, 0,

if y falsifies ;::: m

+1

answers

1/ (m + 1), if y falsifies m answers (5.4) r(y) = m/(m + 1), if y falsifies one answer 1,

if y falsifies no answer .

Any function

r: S

-+

1

m

m+

m+

{0'--1""'--1,1}

is called astate 0/ knowledge in Ulam game over S with m lies/errors. In accordance with this notation, the initial state is the constant function 1 over S. At the other extreme, the constant function 0 is the incompatible state, in which every element of S falsifies m + 1 answers, or more. For every quest ion D ~ S, the positive answer

Dyes : S

-+

{m/(m

+ 1), 1}

is naturally defined by the following stipulation:

(5.5) Dyes(z) = {

1, if z E D 1 - 1/(m + 1), if z E D.

We similarly define the negative answer Dno = YJYes. In other words, for each z E S:

(5.6) Dno(z) =

{

1, if z

Dno by the stipulation

rt. D

1 - 1/(m + 1), if z E D.

The proof of the following proposition is an immediate consequence of (5.3)-(5.6), recalling from (1.2) the definition of Lukasiewicz conjunction:

5.3. OPERATIONS ON STATES OF KNOWLEDGE

107

Proposition 5.2.1 Let t ~ 1 be an integer. For every i = 1, ... ,t, let D i ~ Sand b( i) E {yes, no}. Let r be the state 0/ knowledge arising fw, b(l) b(2) b(t) Jlom the sequence 0/ answers D 1 ,D2 , ... , D t . Then r

= Db(l) 0 1

...

0

Db(t) t,

where 0 is pointwise Lukasiewicz conjunction in the MV-algebra Lm +2 •

o

5.3

Operations on states of knowledge

Onee equipped with pointwise Lukasiewicz eonjunetion, the set KS,m of states of knowledge in Ulam game over the seareh spaee S with m lies/ errors beeomes an abelian monoid with neutral element 1 (the initial state). While Pinoeehio's answers may be false/erroneous, their eonjunetion is all we know ab out the unknown number x. As the number of answers inereases, our state of knowledge beeomes sharper and sharper. To aeeount for the natural order between states of knowledge, let us agree to write

r ' ~ r" (read r ' is sharper than r", or r" is coarser than r ' ) iff r ' (y) ~ r" (y) for all y E S. For every state r E KS,m there is a eoarsest state -,r that is incompatible with r, in the sense that r 0 -,r = O. Naturally, -,r = 1 - r. We ean regard -, as a form of negation, without any fear of eonfusion with (5.5) and (5.6). In partieular, our analysis in Seetion 5.1 (b) does not apply to the -, operation. Using the operations -, and 0 we ean express the natural order between states of knowledge, by writing r 0 -'(7 = 0 instead of r ::; (7. Sinee the operations 0, -', E9 and 1, -', 0 are mutually interdefinable, by a slight abuse of terminology, let us refer to fes,m = (Ks,m, 1, -', 0) as the MV-algebra of states 0/ knowledge in Ulam game over S with m lies. We say that an equation (X1! . .. , X n ) = 'IjJ(X1 , ... ,Xn ) is absolute iff it is valid whenever the variables Xi are replaeed by arbitrary states

Chapter 6 Lattice-theoretical properties In this chapter we study properties that are strongly related to the lattice structure of MV-algebras. We start by considering relations between the ideals of an MV-algebra A and the ideals of the lattice L(A). A stonean ideal of a bounded distributive lattice L is an ideal generated by complemented elements of L. We shall show that the minimal prime lattice ideals of L(A), as weIl as the stonean ideals of L(A), are always ideals of A. An MV-algebra A is called hyperarchimedean iff all its ideals are stonean ideals ofL(A). Hyperarchimedean MV-algebras have many features in common with boolean algebras. For instance, an MV-algebra is hyperarchimedean iff all its prime ideals are maximal. Further, every hyperarchimedean MV-algebra A is semisimple, and so is every homomorphic image of A. We shall describe the most general hyperarchimedean MV-algebra with one generator. We shall also investigate the role of the complemented elements of L(A) in the direct product decompositions of A, and we shall introduce boolean products of MV-algebras as generalizations of finite direct product decompositions. An MV-algebra is said to be complete iff its underlying lattice is closed under infinite sups and infs. The structure of complete MValgebras explicitly depends on the underlying lattice structure, and every complete MV-algebra A is semisimple. We shall also prove that every complete MV-algebra A has a decomposition A = Al X A 2 X A 3 , where Al is complete and atomic, A 2 is apower of [0,1], and A 3 is

111 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

112

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

complete and without linear factors.

6.1

Minimal prime ideals

As usual, by an ideal 0/ a lattice L with 0 and 1 we mean a subset I of L satisfying the following conditions: (IL1) 0 E I; (IL2) If x E land y ~ x then y E I; (IL3) If x, y E I then x V y E I. For every element z E L, the principal ideal ( z 1 is defined by

(6.1)

( z 1= {x E L I x

~

z}.

We say that I is proper iff I =I L. We say that I is prime iff I is proper and for any two x, y E L, if x Ay E I then either x E I or y E I. Every ideal of an MV-algebra A is also an ideal of the underlying lattice L(A). To see that the converse does not hold in general, let z be a nonboolean element of A. Then (z 1 is an ideal of the lattice L(A), but is not an ideal of the MV-algebra A. As a matter offact, by Corollary 1.5.6, an MV-algebra A is a boolean algebra iff every ideal of the lattice L(A) is an ideal of A. Lemma 6.1.1 Let J be an ideal 0/ an MV-algebra A. Then J is a prime ideal 0/ A iff J is a prime ideal 0/ the underlying lattice L(A).

Proo/: Suppose that J is a prime ideal of the MV-algebra A and let x, y E A be such that x A y E J. By hypothesis we can safely assurne x9y E J. Hence (xAY)EB(x9Y) E J. Moreover, from (xAY)EB(x9Y) = (xEB(x9Y))A(yEB(x9Y)) = (xEB(x9y))A(xVy) 2: xA(xVy) = x, we get x E J. Thus, J is a prime ideal of L(A). Conversely, assurne J to be a prime ideal of L(A), and let x, y E A. Since by Proposition 1.1.7, (x 9 y) A (y 9 x) = 0 E J, then we must either have x 9 y E J or y 9 x E J, and, since by assumption J is an ideal of A, then J is a prime ideal in A. 0

6.1. MINIMAL PRIME IDEALS

113

Definition 6.1.2 An ideal K of an MV-algebra A (resp., of L(A)) is said to be minimal prime iff it is a prime ideal of A (resp., of L (A) ), and whenever I ~ K is a prime ideal of A, (resp., ofL(A)), then 1= K. Theorem 6.1.3 Let A be an MV-algebra, and J be a proper ideal of the underlying lattice L(A). For each z E A let

Jz = {x E A I zex ~ J} and K = K(J) = n{Jz

IZ

E

A\J}.

Then K(J) is an ideal of A and K(J) ~ J. Ij, in addition, J is a prime ideal of L(A), then K(J) is a prime ideal of A. Proof: Since 1 E A\J, K is weH defined. We shall prove that K is an ideal of the MV-algebra A. Thivially, 0 E K. If t ::; u E K, then t E K. Indeed, for each z E A\J from z e u ~ J we get a fortiori, z e t ~ J (because, z e t ;::: z e u). Thus t E K, as required. Assurne now x, y E K, with the intent of proving x EIl y E K. Pick an element z E A\J. Then from x E K we get z ex E A\J, and hence y E K implies (z e x) e y ~ J. Thus,

z e (x EIl y) = z 0 (....,x 0 .y) = (z e x)

e y ~ J,

whence x EIl y E K. We have thus shown that K is an ideal of A. In order to prove that K ~ J, it is enough to observe that Z e OE J; thus, whenever Z ~ J, we must have z ~ K. Claim: For each Z E A, if x 1\ y E J z , then either x E Jz or y E Jz. As a matter of fact, we have the identities

(z 0 .x) V (z 0 .y) = z 0 (....,x V ....,y) = z e (x

1\

y)

~

Z

=

J,

and since J is an ideal of L(A), we must either have z e x ~ J or z e y ~ J, as required to settle our claim. To conclude the proof, assurne J to be a prime ideal of L(A). Assume x ~ K and y ~ K. Then there exist elements s, t E A \ J such that x ~ Js and y ~ Jt . By our assumption about J, s 1\ t ~ J. If x 1\ Y E K (absurdum hypothesis) then x 1\ y E JSl\t and hence, by our claim, we must either have x E JSl\t or y E JSl\t. From the inclusion JSl\t ~ Js n Jt we obtain a contradiction in either case, thus showing that x 1\ y ~ K. We conclude that K ~ J::/: A is a prime ideal of A. 0

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

114

Corollary 6.1.4 Let A be an MV-algebra and L(A) its underlying lattice. Then the set of minimal prime ideals of A coincides with the set of minimal prime ideals ofL(A). Proof: This is an immediate consequence of Theorem 6.1.3, together with Lemma 6.1.1. D

Let us recall that a dual ideal, or filter, of a lattice L is a subset F of L satisfying the following conditions: (Fl) 1 E F; (F2) If xE Fand x

~

y, then y E F

(F3) If x E Fand y E F, then x A y E F. Theorem 6.1.5 Let J be a minimal prime ideal of an MV-algebra A. Then for every s E J there exists t E A \ J such that t A s = O. Proof" By Corollary 6.1.4, J is a minimal prime ideal of L(A). Let F = {y E A I there is x E A \ J such that sAx ~ y}. Then F is a filter of L(A) and s E F. Moreover, for every x E A\J we have x E F, in symbols,

(6.2)

A\J ~ F.

Claim: If 0 fj. F then there exists a prime ideal P of L(A) such that

pnF=0.

As a matter of fact, by our standing assumption, the family of all ideals of L(A) that are disjoint from F is nonempty. Applying Zorn's lemma, let P be maximal in this family.. Then P will be a prime ideal of L(A). For otherwise, letting a, b fj. P and a A bE P, by our assumption about P there are p, q E P such that p V a E Fand q V bE F. Then (p V a) A (q V b) E F. On the other hand, by distributivity,

(p V a)

A

(q

V

b)

= (p A q) V (p A b) V (a A q) V (a A b)

E P,

thus contradicting the disjointness of P and F. In conclusion, P is prime and the claim is settled. By (6.2) we now get P ~ J and s E J\P, thus contradicting the fact that J is a minimal prime ideal of L(A). It follows that 0 E F, whence there exists t E A \ J such that s A t = 0, as required. D

6.2. STONEAN IDEALS AND ARCHIMEDEAN ELEMENTS

6.2

115

Stonean ideals and archimedean elements

Definition 6.2.1 Let L be a lattice with 0 and 1. An ideal I of L is said to be stonean Hf for every x E I there is a boolean element Z E InB(L) such that x ~ z.

Stated otherwise, an ideal I of L is stonean iff it is generated by the ideal I n B (L) in the boolean algebra B (L). For every x E L, the ideal (x] is stonean iff x E B (L ). The following is a generalization of Corollary 1.5.6. Corollary 6.2.2 For any MV-algebra A, every stonean ideal of the underlying lattice L(A) is automatically an ideal of A. Proof: Let J be a stonean ideal of L(A). We have only to show that J is closed under the EB operation. Let x, y E J. Then there are s, tE JnB(A), such that x ~ sand y ~ t. Since by Theorem 1.5.3(iv), x EB Y ~ s EB t = sV t E J n B(A), we obtain x EB y E J. 0 Definition 6.2.3 An element x of an MV-algebra A is said to be archimedean iff there is an integer n 2: 1 such that x EB ... EB x (n times) is boolean. Notation: Throughout this chapter, following the notation of (2.7) we shall write n.x

as an abbreviation of x EB ... EB x (n times).

Corollary 6.2.4 Por every element x of an MV-algebra A, the following conditions are equivalent:

(i) There is an integer n 2: 1 such that ...,x V n.x (ii) There is an integer n 2: 1 such that n.x (iii) x is archimedean; (iv) (x) is a stonean ideal ofL(A).

=

(n

= 1;

+ l).x;

116

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

Proof: (i) => (ii): If 1 = -'x V n.x = -,(x E9 n.x) E9 n.x then n.x ~ (n + l).x = x E9 n.x ~ n.x. (ii) => (iii): If n.x = (n + l).x, then by induction we get n.x = (n + k).x, for all integers k ~ O. In particular, n.x = 2n.x and by Theorem 1.5.3(iv), n.x E B(A). (iii) => (iv): Let no ~ 1 be an integer such that no.x E B(A). By Theorem 1.5.3(iv), no.x = 2no.x = kno.x for all k ~ 2. Assume Y E (x). Then Y ~ n.x for some n, whence a fortiori, for suitably large k, we get Y ~ kno·x E B(A) n (x). Thus (x) is a stonean ideal of L(A). (iv) => (i): By assumption, since x E (x), there is z E B(A) n (x) such that x ~ z. Since z E (x), there is an integer n ~ 1 such that z ~ n.x, whence, by Theorem 1.5.3(vi), 1 = -,z E9 n.x = -,z V n.x ~ -,x V n.x ~ 1. 0

6.3

Hyperarchimedean algebras

Definition 6.3.1 An MV-algebra A is said to be hyperarchimedean iff all its elements are archimedean. Equivalently, by Corollary 6.2.4(ii), for every x E A there is an integer n ~ 1 such that n.x = (n+1).x; one then immediately sees that boolean algebras, finite MV-algebras and simple MV-algebras are examples of hyperarchimedean MV-algebras. If the MV-algebra A is a chain, then B(A) is the two-element set {O, I}. In this case, a nonzero element x is archimedean iff 1 = n.x for some integer n ~ 1. Then by Theorem 3.5.1, an MV-chain is hyperarchimedean iff it is isomorphie to a subalgebra of [0,1]. While the class of hyperarchimedean MV-algebras is closed und er subalgebras, finite products and homomorphic images, this class is not closed under infinite products. As a matter of fact, let

(6.3)

A

=

II

Ln·

n~2

Then each Ln is hyperarchimedean, but the product MV-algebra A is not. To see this, let f : {2, 3, 4, ... } - t Un>2 Ln be defined by f(n) = l/(n - 1). From 0< n.f(n + 2) = nf(n +-2) = n/(n + 1) < 1, we get

6.3. HYPERARCHIMEDEAN ALGEBRAS

117

n.f f/. B(A), for each n = 0,1,2, ... , whence f is a nonarchimedean element of A, and A is not hyperarchimedean.

Theorem 6.3.2 For any MV-algebra A the following conditions are equivalent:

(i) A is hyperarchimedean; (ii) Every ideal of A is a stonean ideal ofL(A); (iii) Every prime ideal of A is maximal; (iv) Every prime ideal of A is minimal; (v) Every ideal of A is an intersection of the maximal ideals containing it; (vi) For every ideal J of A, AI J is semisimple. Proof: (i):=;. (ii): Let J be an ideal of A, and x E J. By Corollary 6.2.4(iii) there is an integer n > 0 such that n.x is a boolean element. Since x ~ n.x E J, J is a stonean ideal of L(A). (ii) :=;. (iii): Let P be a prime ideal in A and P ~ J for some ideal J in A. Let Po = pnB(A), J o = JnB(A). Suppose x E J\P (absurdum hypothesis ). Since by hypothesis J is stonean, there is z E Jo such that x ~ z; it follows that z f/. Po. Since P is prime, it is easy to see that Po is a maximal ideal in the boolean algebra B(A). Since Jo strictly contains Po, we must have 1 E Jo, whence J 2 J o is not a proper ideal, a contradiction showing that P is maximal. (iii) <=? (iv): Thivial. (iv) :=;. (i): Let 0 =1= s E A, and J.L = {x E A I x 1\ s = O}. Then J.L is an ideal of A. As a matter of fact, skipping all trivialities, whenever x 1\ s = 0 = y 1\ s, then (x EB y) 1\ s = 0; this is easily verified on all MVchains, and hence by the subdirect representation theorem, it holds in general. Let K be the ideal generated by J.L and s. By (1.9) we have K

= {x

E A

Ix

~ n.s EB a, for some n

= 1,2, ... and a E J.L}.

Assurne K to be a proper ideal (absurdum hypothesis). Then, by Corollary1.2.14 there is a prime ideal J in A containing K. Since

118

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

by hypothesis, J is minimal, an application of Theorem 6.1.5 yields an element tE A\J such that s A t = O. Since J;2 K ;2 J1.., then t J1.., a contradietion. We have shown that 1 E K. Thus, there is an integer n ;::: 1 and an element a E J1.. such that n.sEBa = 1. From aAs = 0 and ->n.S ~ a it follows that -m.sAs = 0, whence 1 = n.sV-,s. By Corollary 6.2.4(i), s is an archimedean element of A, and A is hyperarchimedean. (iii) {::} (v): This is a consequence of Corollary 1.2.14. (v) {::} (vi): By Lemma 3.6.6. 0

tt

By Corollary 3.6.8, since each hyperarchimedean MV-algebra A is semisimple, there is a compact Hausdorff space X such that A is isomorphie to a separating subalgebra of Cont(X).

Proposition 6.3.3 Let X be a compact Hausdorff space. A separating subalgebra A of Cont(X) is hyperarchimedean iff ZU) = f-l( {O}) is an open subset of X for each f E A.

Proof" Suppose that A is hyperarchimedean, and for each x EX, let Ox denote the ideal given by all functions in A that vanish over some open neighbourhood of x. The fact that A separates points implies that VOx = {x}, and then, from Theorem 3.4.3(iii) and Theorem 6.3.2(vi) it follows that Ox = Jvox = J x' Therefore, whenever f E A vanishes in some point x E X, then f also vanishes over an open neighbourhood of x. This property clearly implies that ZU) is an open subset of X for each f E A. To prove the converse, we need the following Claim: Let J be an ideal of A, and let f E A. If f vanishes over so me open set U;2 VJ , then f E J.

As a matter of fact, since VJ n (X \ U) = 0, for each x E X \ U there is a function gx E J such that gx(x) > O. Since X \ U is a compact subset of X, arguing as in the proof of Theorem 3.4.3(ii), we can find a function 9 E J such that g(x) = 1 for each x E X \ U. Since f(x) = 0 ~ g(x), for all x E U, and f(x) ~ 1 = g(x), for all x E X \ U, then f ~ g. We conclude that f E J, as required to settle our claim. Suppose now that ZU) is open for each f E A, and let J be a proper ideal of A. If f E JvJ , then VJ ~ ZU), and by the above

6.3. HYPERARCHIMEDEAN ALGEBRAS

119

claim, I E J. Therefore J = JvJ , and again by Theorem 3.4.3(iii) and Theorem 6.3.2(vi) we can conclude that A is hyperarchimedean. 0 We shall now characterize hyperarchimedean MV-algebras with one generator. In the light ofTheorem 3.6.7 and Proposition 3.1.5, we can safely focus attention on MV-algebras of rest riet ions of McNaughton functions, as follows:

=I 0 be a closed subset 01 the real interval [0, 1], and let A be the subalgebra 01 Cont(X) given by the restrietions to X 01 the lunctions in Freel. Then A is hyperarchimedean iff every rational point 01 X is isolated.

Corollary 6.3.4 Let X

Proof" By Corollary 3.2.8, Freel coincides with the MV-algebra of all one-variable McNaughton functions. Suppose that A is hyperarchimedean and let r be a rational point in X. We can easily define a McNaughton function Ir: [0, 1] ~ [0,1] such that Z(fr) = {r}. Then from Proposition 6.3.3, we obtain that {r} = Z (fr) n X is open in X, i.e., r is isolated in X. Conversely, suppose that all rational points in X are isolated, and let I: [0, 1] ~ [0,1] be a McNaughton function. Clearly Z(flx) = Z(f) n X. Let z E Z(f) n X. If z is rational, then {z} is an open neighbourhood of z (in the topology of X) that is contained in Z(flx). Suppose that z is irrational. Since I is aMeN aughton function, there is an open interval U such that z E U ~ Z(f). Then U n X is an open neighbourhood of z in X and is contained in Z(flx). This shows that Z(flx) is open in X, whence by Proposition 6.3.3, A is hyperarchimedean. 0

Let X be a compact Hausdorff space. Following tradition, we say that a set S ~ X is dopen iff S is simultaneously open and closed. and X are always clopen subsets of X. The clopen subsets of X equipped with union, intersection and complement, form a boolean algebra, denoted clop(X). On the other hand, it is easy to see that

o

= {I E Cont(X) I for all x E X, I(x) E {O, I}}. Therefore the correspondence I 1----+ Z (f) defines an iso morph ism from B(Cont(X))

B(Cont(X)) onto clop(X). More generally, let U be a clopen sub set of

120

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

X and A be a separating subalgebra of Cont(X). Since U is compact, for each x f/. U there is Ix E A such that Ix(x) = 1 and U ~ Z(fx). Since X \ U is also compact, we can find I E A such that I(t) > ~ for each tEX \ U and Z(f) = U. Hence 9 = 2.f E B(A) and Z(g) = U. In condusion, we have proved that the correspondence 9 ~ Z(g) defines an isomorphism 01 B(A) onto dop(X). In particular, for each separating sub algebra A of Cont(X), we have B(A) ~B(Cont(X)). Let X be a compact Hausdorff space and A be a separating subalgebra of Cont(X). If A is hyperarchimedean then, by Proposition 6.3.3, the complements of the zero sets of functions in A are dopen and, by the remark following Theorem 3.4.3, they form a basis of the topology of X. Compact Hausdorff spaces having a basis of dopen sets are called boolean spaces . For every boolean algebra BIet

X(B) be the set of maximal ideals of B, equipped with the topology having as an open basis the sets ofthe form O"B(a) = {P E X(B) la f/. P}, where a ranges over elements of B. Then X(B) is a boolean space. Furthermore, Stone's celebrated theorem states that B ~ dop(X(B)), and X(B) is uniquely determined, up to homeomorphism. Since B(Cont(X)) ~ clop(X), we conclude that a Hausdorff compact space X is a boolean space iffB(Cont(X)) is a separating subalgebra oICont(X). We have proved: Corollary 6.3.5 Let X be a compact Hausdorff space. Then the MValgebra Cont(X) has a hyperarchimedean separating subalgebra A iff X is a boolean space. In this case, X is homeomorphic to X(B(A)). 0 It is worthwhile to point out that the fact that X is a boolean space need not imply that Cont(X) is a hyperarchimedean MV-algebra, as the following example shows.

Example: Let X be Cantor's ternary subset of [0, 1]. It is well known that X, with the topology inherited from [0, 1] is a boolean space having no isolated points. Since X contains rational points, by Corollary 6.3.4,

6.4. DlRECT PRODUCTS

121

the separating subalgebra of Cont(X) formed by the restrietion of McNaughton functions to X is not hyperarchimedean, whence Cont(X) is not hyperarchimedean.

6.4

Direct products

Let A be an MV-algebra. For each z E A let the functions and hz : A ~ A be defined by

(6.4)

-'z :

A

~

A

hz(x) = z A x and -'zX = z A -,x.

Proposition 6.4.1 For each nonzero element bE B(A), (( b], EB, -'b, 0) is an MV-algebra and hb is a homomorphism of A onto (b] with Ker(h b ) = (-,b].

Proof: By Corollary 1.5.6, (b] is an ideal of A, and hence it is closed under the restrietion of the operation EB over (b]. We shall now show that (b] is an MV-algebra. We first note that (b] is closed under the operation -'b, and that for all x E (b], -'b-'bX = x. Further, by Theorem 1.5.3(vi), for every x E (b] we have x EB -'bO = x EB b = x V b = b. To establish the identity -'b(X EB -'bY) EB x = -'b(-'bX EB y) EB y, we first note that for any y E (b], b EB y = b V Y = b. It follows that -'b(-'bX EB y) EB y = -'b((b A -,x) EB y) EB y

= -'b ( (b EB y)

A (-,x EB y)) EB y

= (b A -,(b A (-,x EB y))) EB y = ((b A -,b) V (b A -,( -,x EB y))) EB y = (b EB y) A (-, (-,x EB y) EB y) = b A (-, (-,x EB y) EB y). By symmetry,

-'b(X EB -'bY) EB x = bA (-,(x EB -,y) EB x) = bA (-,( -,x EB y) EB y) = -'b ( -'bX

EB y) EB y,

as required. The remaining verifications to establish that (b] is an MValgebra, are all trivial. In order to prove that hb is a homomorphism,

122

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

for all x, y E A we have (x 1\ b) EB (y 1\ b) = ((x 1\ b) EB y) 1\ ((x 1\ b) EB b). Since, by Theorem 1.5.3(vi), (x 1\ b) EB b = (x 1\ b) Vb = b, we get

(x 1\ b) EB (y 1\ b) = (x EB y) We conclude that hb(x EB y) trivial. 0

1\

(b EB y)

1\

b = (x EB y) 1\ b.

= hb(x) EB hb(y). The rest of the proof is

°

Definition 6.4.2 For any partially ordered set X with minimum element 0, by an atom of X we mean an element a E X such that a > and whenever x E X and x ~ athen either x = or x = a.

°

The following is an immediate consequence of the definition of (b J:

°

< b E B(A) we have (i) The MV-algebras ( bJ and AI (-,b] are isomorphie; (ii) (b] is a subalgebra of A iff b = 1 iJJ (b] = A; (iii) B(( b]) = (b J n B(A). If in addition, (b J is a ehain, then b is an atom of the boolean algebra B(A). 0

Proposition 6.4.3 For every MV-algebra A and

°

Remark: For each < a E A, upon defining x EBa Y = (x EB y) 1\ a and -'aX = a 8 -,x, it follows that (( a], EB a , -'a, 0) is an MV-algebra. However, if a is not a boolean element of A, then in general (aJ is not a homomorphic image of A. For instance, let n 2: 3 and < k < n - 1. Then k/(n-1) is an element ofthe Lukasiewiez chain Ln and (kl(n-1) 1 is isomorphie to L k +1' Since k + 1 < n and Ln is simple, there is no homomorphism of Ln onto (k/(n - 1)]. On the other hand, the existence of a homomorphism of A onto ( a] need not imply that a is a boolean element of A. As a matter of fact, for each < a E [0, 1J, the map x ~ ax is an isomorphism of [0,1] onto (a]. However, a is not a boolean element of [0,1], unless a=l.

°

°

Notation: Given a nonempty family {ai hEl of elements of an MValgebra A, we write ViEl ai = 1 iff 1 is the only upper bound of the family.

6.4. DIRECT PRODUCTS

123

Lemma 6.4.4 Let {AihEl be a nonempty family of MV-algebras and let P = IIiEl Ai. Then there is a set {8i I i E I} ~ B (P) satisfying the following eonditions:

(i) ViEl 8i

(ii) 8i

/\

8j

= 1; = 0, whenever i

=1=

j;

(iii) eaeh Ai is isomorphie to (8i ]. Proof: For each i E I, let 8i : 1---+ UiEl Ai be defined by 8.( ") ~ J

= { 1 E Ai if

j 0 E Ai if j

=i

=1=

i.

Then 8i E B(P), 8i /\8j = 0, for all i =1= j, and ViEl 8i = 1. Let 7l"( P ---+ Ai be the canonical projection. Then the kernel of h6i : P ---+ (8i ] coincides with the kernel of 7l"i. As a matter of fact, by Corollary 6.4.1 we can write Ker(h 6J = (,8d = {f E P I f(i) = O} = Ker(7l"i)· Thus, by Lemma 1.2.7, Ai is isomorphie to (bd. 0 The above lemma has the following partial converse:

Lemma 6.4.5 Let A be an MV-algebra. Let bl , ... ,bk, k ments in B (A) sueh that

(i) bl

V ... V

~

2 be ele-

bk = 1, and

(ii) bi /\ bj = 0 for i Then A

~

(bd x ...

X

=1=

j, i, j

= 1, ... , k.

(bk]'

nf=l

Proof" From (i) we immediately get (,bi] = {O}. Hence by Theorem 1.3.2 and Proposition 6.4.1, the function h: A ---+ (bI] X ... x (bk] given by h(a) = (a /\ bl , ... , a /\ bk), for each a E A, is an embedding. Since by (ii), h(al V ... Vak) = (al,"" ak), then h is also surjective, whence h is an isomorphism. 0 Definition 6.4.6 An MV-algebra A is called direetly indeeomposable iff Ais nontrivial, and whenever we can write A Al X A 2 then either Al or A 2 is trivial. f"'oo.I

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CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

As an immediate consequence of Lemmas 6.4.4 and 6.4.5 we have:

Theorem 6.4.7 An MV-algebra A is directly indecomposable if and onlyifB(A)={O,l}.O Prom Proposition 6.4.3(iii) and the above theorem we obtain that for each atom b of B(A), the MV-algebra (b] ~ AI( -,b] is directly indecomposable. This result can be generalized as follows: first of all, let us say that an ideal J of an MV-algebra A is a maximal stonean ideal iff J is a proper stonean ideal of A, and for each stonean ideal I of A, I =I J and J ~ I imply I = A. In other words, J n B(A) is a prime ideal of B (A).

Corollary 6.4.8 Let A be a nontrivial MV-algebra. For each maximal stonean ideal J of A, the quotient algebra AI J is directly indecomposable. Proof: Let x E A be such that xl JE B(AI J). Then x A -,x E J, and there is z E J n B(A) such that x A -,x ~ z. Letting y = x V z, we get y A -,y = (x A -,x A -,z) V (z A -,x A -,z) = 0. It follows that y E B(A), and since J n B(A) is a prime (= maximal) ideal of B(A), then either y E J or -,y E J. If y E J, then x E J, because x ~ y. Suppose y ~ J. Then -,y E J, and -,x V z = -,y V z E J. Hence -,x E J. 0

6.5

Boolean products of MV-algebras

As shown by Lemmas 6.4.4 and 6.4.5, boolean elements have an important role in direct product decompositions of MV-algebras. We will return to this topic in the next section, when considering complete MV-algebras. Now we are going to consider a special kind of subdirect product, where the index set is equipped with a boolean topology.

Definition 6.5.1 A weak boolean product of a family {AX}XEX of MValgebras (X =I 0) is a subdirect product A of the given family, in such a way that X can be endowed with a boolean (i.e., totally disconnected, compact Hausdorff) topology having the following two properties:

6.5. BOOLEAN PRODUCTS OF MV-ALGEBRAS

(i) For all J,g E A, the set [J open in Xj

= g]

=def

{x E X If(x)

125

= g(x)}

(ii) Whenever Z is a dopen (dosed-and-open) subset of X and Athen Jlz U glx\z E A.

is

J, gE

Replacing condition (i) by: (i*) For all

J, 9 E

A, the set

[J = g] is dopen,

we obtain the notion of a boolean product. Recalling the properties of the distance function, since a = b is equivalent to d(a, b) = 0, the above conditions (i) and (i*) can be replaced, respectively, by: (i') If J E A, then

[J = 0] is open in X

and (i*') If J E A, then

[J

=

0] is dopen in X.

In the following, when dealing with a weak boolean product A of a family {AX}XEX, the dependence of A on the topology of X shall be tacitly understood. As shown by the following example, boolean products are a generalization of finite direct products.

Example: Let {AX}XEX, for X =f:. 0 be a family of nontrivial MValgebras, and let A = IIxEx A x . Then A is a weak boolean product of the family {AX}XEX if, and only if, X is finite. In this latter case, A is indeed a boolean productj further, the discrete topology on X is the only possible topology making A into a boolean product of the family {AX}XEX' As a matter offact, assuming X to be finite, then conditions (i*') and (ii) are trivially satisfied upon equipping X with the discrete topology. A moment's reflection shows that no other topology on X can make A into a boolean product of the {AX}XEX' Conversely, assume X to be infinite, with the intent of proving that A cannot be made into a

126

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

weak boolean product of the {AX}XEX' For each z E X let the function fz: X ---. UXEX A x be defined by stipulating that, for each x E X, if x if x

= z, # z.

Since f E A and [fz = 0] = {z}, then, for condition (i) to hold it is necessary to endow X with the discrete topology. While this topology is Hausdorff and has a basis of dopen sets, it fails to make X into a compact space. The following theorem should be compared with Lemmas 6.4.4 and 6.4.5, together with the above example. Recall the definition of X( C) for any boolean algebra C. Theorem 6.5.2 Let A be a weak boolean produet of a family {A x }XEX, X #0, of nontrivial MV-algebras. Let C be defined by

(6.5)

C = {g E AI g(x) E {Ox, Ix} for eaeh x EX}.

Then C is a subalgebra of B(A) and we have:

(i) The eorrespondenee x ~ Qx = {g E morphism from X onto X( C);

CI g(x) =

O} is a homeo-

(ii) For eaeh x E X, A x is isomorphie to AI(Qx); (iii) C eoineides with B(A) iJJ all algebras A x are direetly indeeomposable. Conversely, if A is a nontrivial MV-algebra and C is a subalgebra of B(A), then A is isomorphie to a weak boolean produet of the family {AI (Q) }QEX(C). Proof: Suppose first that A is a boolean product of the family {A x } xEX, and let C be as in (6.5). Then C is a subalgebra of B(A), and by Definition 6.5.1, C ~ B(X); arguing as in Theorem 3.4.3 one easily sees that the map x ~ Qx is a one-one correspondence of X onto X(C). As a matter of fact, this correspondence is continuous, because for each x E X and 9 E C we have Qx E ac(g) iff x E X \ [g = 0] = [.g = 0],

6.5. BOOLEAN PRODUCTS OF MV-ALGEBRAS

127

the latter being open by condition (i) in Definition 6.5.1. Hence, inverse images of basic open sets of X(C) are open in X. Now (i) follows from the weIl known fact that continuous bijections between compact Hausdorff spaces are homeomorphisms. Arguing as in the proof of Theorem 1.3.2, from the assumption that A is a subdirect product of the family {AX}XEX, letting for each x E X, Px = {I E AI I(x) = O} = Ker(7rx ), it is easy to see that A x ~ A/Px. Hence to prove (ii) we need to show that Px = (Qx) for each x E X. Suppose 1 E Px. Then x E [/ = 0], and since x ~ Px is a homeomorphism from X onto X( C), there is an element 9 E C such that x E [g = 1] = [-,g = 0] ~ [I = 0l Therefore, -,g E Px n C = Qx and 1 ~ -'g. Hence Px = (Qx), as required. If C = B(A), for each x E X, Px = (Qx) is a maximal stonean ideal of A, and by Corollary 6.4.8, A/(Qx) is directly indecomposable. On the other hand, iffor some x E X, A x is not directly indecomposable then Qx cannot be a prime ideal of B(A). Since Qx is a prime ideal of C, the latter must be a proper subalgebra of B(A). Hence (iii) holds. Conversely, assurne that C is a subalgebra of B(A). Then, in the light of Theorem 1.3.2, in order to prove that A is a subdirect product of the family {(Q) }QEX(C), it suffices to prove the following:

Claim. The intersection of all the ideals of A that are generated by prime ideals of C coincides with the ideal {O}. Let 0 =1= a E A. Applying Corollary 1.2.14 to the ideal {O}, we get a prime ideal P of A such that a fj. P. Since P n C is a prime ideal of C, to complete the proof it suffices to show that a fj. Q = (P n C). Suppose that a E Q (absurdum hypothesis). Then there is c E P n C such that a ~ c. Moreover, a 1\ -,c = 0 E P, and since a fj. P, by Lemma 6.1.1 we obtain -,c E P. Hence, 1 = cE9 -,c E P, and P = A, a contradiction. This proves the claim. To simplify the notation, we can safely identify A with its corresponding subalgebra of TIQEX(C) A/(Q), and regard the elements of A as functions I:X -+ UQEx(c)A/(Q). Suppose 1 E A, and let Q E [I = 0]. Then / E (Q), and there is gE Q such that 1 ~ g. It follows that

Q E ac(-,g) = [g = 0]

~

[J = 0],

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CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

whence condition (i) in Definition 6.5.1 holds. Finally, let Z be a dopen subset of X. Then there is 9 E C such that Z = ac(g), and since for all fand h in A, flz U hlx\z = (g 1\ 1) V (-,g 1\ h) E A, we condude that (ii) in Definition 6.5.1 also holds. 0 Corollary 6.5.3 Eaeh nontrivial MV-algebra is isomorphie to a weak boolean produet of direetly indeeomposable MV-algebras. 0

Let A be an MV-algebra. Since AI J is an MV-chain iff J is a prime ideal, and MV-chains are directly indecomposable, we obtain: Corollary 6.5.4 A nontrivial MV-algebra A is a weak boolean produet of MV-ehains if and only if eaeh maximal stonean ideal of A is a prime ideal of A. 0

Particular cases of MV-chains are given by simple MV-algebras, i.e., the subalgebras of [0, 1]. Since for any MV-algebra A, AI J is simple if and only if J is maximal ideal, we obtain: Corollary 6.5.5 An MV-algebra A is a weak boolean produet of simple algebras i.tJ eaeh maximal stonean ideal of A is a maximal ideal of A.

o

Suppose that A is a weak boolean product of simple MV-algebras. and let a E A and Q E X(B(A» be such that Q E [a =1= 0]. Then a does not belong to the ideal (Q) generated by Q in A, and since by Corollary 6.5.5 (Q) is a maximal ideal of A, there are e E (Q) and n E N such that c EB na = 1, i.e., -,na ::; c. Hence there is b E Q such that -,na ::; b, whence Q E [b = 0] ~ [a =1= 0]. Therefore [a = 0] is dopen for each a E A. We have proved that weak boolean products of simple MV-algebras are automatically boolean products. On the other hand, if P is a prime ideal of an MV-algebra A, then (P n B(A») is a maximal stonean ideal contained in P. Therefore, in the light of Theorem 6.3.2 we conclude that each maximal stonean ideal of A is a maximal ideal of A i.tJ A is hyperarehimedean. From the above remarks and Corollary 6.5.5, we get the following characterization of hyperarchimedean MV-algebras:

129

6.6. COMPLETENESS

Corollary 6.5.6 A nontrivial MV-algebra A is isomorphie to a boolean produet of simple MV-algebras iJJ A is hyperarehimedean. 0 The example given at the end of Section 6.3 shows that in general, Cont(X), for a boolean space X, is not a boolean product of subalgebras of [0,1].

6.6

Completeness

By definition, a lattice L is complete iff every subset {Xi I i E I} of L has a supremum and an infimum, which we respectively denote by

VXi ,

( or

V{Xi I i E I} )

AXi ,

(or

A{Xi I i EI}).

iEI

and

iEI

Any complete lattice has a minimum element 0 = V 0 and a maximum element 1 = 1\ 0. Definition 6.6.1 We say that an MV-algebra A is eomplete iff its underlying lattice L(A) is complete. We say that A is a-eomplete iff suprema and infima exist for all finite or denumerable subsets in L(A). By Theorem 3.5.1, the only complete and simple MV-algebras are [0, 1] and the finite chains Ln. Proposition 6.6.2 Every a-complete MV-algebra A (whence, a fortiori, every eomplete MV-algebra) is semisimple. Proof: Assuming X E Rad(A), we shall show that of fact, by Proposition 3.6.4 we have

(6.6)

n.x ~ -'X,

Le.,

X 0 n.x

X

= o. As a matter

= O.

Let s = VnEN(n.X). Then for all n = 0,1, ... , we have (n + l).x ~ s. From the monotonicity ofthe 0 operation it follows that (n+1).xex ~ sex. From (6.6) we now get n.x

= n.x 1\ -,x =

(n.x ED x) ex

= (n + l).x e X ~

sex.

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CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

Therefore, s :::; required. 0

sex:::;

s, whence, 0 = se (s e

x)

Lemma 6.6.3 Let A be a complete MV-algebra. Let Then

= s 1\ x = {Xi

x,

as

li E I} ~ A.

1\ Xi = -, V -'Xi

(6.7)

iEI

iEI

and

VXi = -, 1\ -'Xi •

(6.8)

iEI

iEI

Proof" Prom the elementary properties of the natural order relation in A.D

Lemma 6.6.4 Let A be a complete MV-algebra. Let {Xi li E I} ~ A. Then for each X E A, the following generalized distributive laws hold:

(6.9)

X 1\

VXi = V(x 1\ Xi)

iEI

iEI

and

(6.10)

X V

1\ Xi = 1\ (X V Xi).

iEI

iEI

Proof" We first prove

(6.11)

X8VXi=V(x8xi)' iEI iEI

To this purpose, let a = ViEl Xi' Since Xi :::; a, for every i E I, X 8 Xi :::; X 8 a. Let us assume that for each i EI, X 8 Xi :::; z. By Lemma 1.1.4(iii), Xi :::; -'x EB z for each i E I. Hence a :::; -'x EB z, and by the same lemma, we get X 8 a :::; z. Thus, X 8 a = ViEI(X 8 Xi), which settles (6.11). Now by (6.8), for every i E I we can write -,a :::; -'Xi; hence, by (6.11) X 1\

a

=

(-,a EB x) 8 a

=

V(( -,a EB x) 8 Xi)

iEI

6.6. COMPLETENESS

iEI

131

iEI

Prom the trivial inequality ViEI(X 1\ Xi) :5 X 1\ a we finally obtain (6.9). To complete the proof, it is sufficient to note that, by Lemma 6.6.3, the identities (6.9) and (6.10) are equivalent. 0 Corollary 6.6.5 Let A be a complete MV-algebra. Then

(i) B(A) is a complete boolean algebra. As a matter of fact, for every set {bi li E I} ~ B(A) we have (6.12)

Vb E B(A) i

iEI

and

(6.13)

1\ bi E B(A); iEI

(ii) For every b E B(A), letting (b land h b : A -+ (b 1 be as in (6.1) and (6.4), it follows that (b 1 is a complete MV-algebra, and h b preserves arbitrary infima and suprema. In more detail,

and

Proof" (i) Let ,x

=

X

1\ ,bi iEI

=

ViEl

bio By (6.7) and (6.9), we have

and ,x 1\ x

=

V(,x 1\ bi )

iEI

::;

V(,bi 1\ bi ) = O. iEI

Then by Theorem 1.5.3(iii), x is a boolean element of A; this settles (6.12). The proof of (6.13) is similar. (ii) Since (b 1 is a complete MV-algebra, (6.14) and (6.15) are an immediate consequence of the definition of hb , in the light of (6.9) and (6.10). 0 Using Lemma 6.6.4 an easy adaptation of the proof of Lemma 6.4.5 yields

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CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

Lemma 6.6.6 Let A be a eomplete MV-algebra. Let the set {bi I i E I} ~ B(A) satisfy the following eonditions: ViEl bi = 1, and bi 1\ bj = 0 whenever i =J j. Then A is isomorphie to the direct produet MV-algebra

I1{(bd

li E I}.

0

From Lemmas 6.4.4 and 6.6.4 we get that for every complete MValgebra A, there is a one-one correspondence between the direct product decompositions of A and the sets {bi I i E I} ~ B(A) such that ViEl bi = 1 and bi 1\ bj = 0 whenever i =J j. More generally, given an infinite cardinal K and a K-complete MV-algebra A (as defined by a natural extension of 6.6.1), an easy adaptation of the proofs of Lemmas 6.4.4 and 6.6.4 yields a one-one correspondence between direct product decompositions of A into K many factors, and sets {ba I Q' E K} ~ B(A) such that VaEIt ba = 1 and ba 1\ bß = 0 whenever Q' =J ß. The crucial point is that in Lemma 6.6.4 the supremum on the right hand side exists Hf so does the supremum on the left hand side, and if this is the case the suprema coincide.

I i E I} be a family of MV-algebras. Then the produet P = I1iEI Ai is a eomplete MV-algebra iff so is eaeh Ai·

Lemma 6.6.7 Let {Ai

Proof: It is easy to check that the two lattices L(P) and I1 iEI L(A i ) coincide. Clearly, the direct product of complete lattices is a complete lattice. Thus, if each Ai is complete, then so is P. The converse is an immediate consequence of Corollary 6.6.5(ii), together with Lemma 6.4.4. 0

6.7

Atoms and Pseudocomplements

Definition 6.7.1 By an atom of an MV-algebra A we mean an atom of the underlying lattice L(A). We say that A is atomie Hf for each o =J x E A there is an atom a E A with a ~ x. We say that A is atomless iff no element of A is an atom. Examples. For each n ;::: 2, the element l/(n - 1) is an atom of the MV-chain Ln. By Proposition 3.5.3, all infinite subalgebras of [0, 1] are atomless. By Theorem 3.5.1 and Proposition 3.5.3, up to isomorphism,

6.7. ATOMS AND PSEUDOCOMPLEMENTS

133

simple, complete, atomic MV-algebras coincide with the MV-chains Ln, where n ~ 2. Further, [0,1] is the only simple, complete and atomless MV-algebra. In the rest of this section, we shall study algebras of the form ( b], when bis an atom of the boolean algebra B(A). Let A be a complete MV-algebra. For each z E A, recall that, by definition, J/. = {x E A I x A z = O}. It is not hard to see that Jz 1. is an ideal of A. (To this purpose, using the subdirect representation theorem one simply notes that for all p, q, rE A if p A q = 0 then pAr =pA(qEBr)). Let zoO E A be defined by zoO = VJz 1. = V{x I x A z = O}. Then z* is the pseudoeomplement of z, in the sense that for any x E A

(6.16)

xAz=O iff x$z*.

Lemma 6.7.2 Let A be a eomplete MV-algebra. Then the lattiee L(A) is pseudocomplemented, in the sense that eaeh z E A has its pseudoeomplement z". M oreover, for each z E A the pseudocomplement z" is a boolean element of A.

Proof: By Lemma 6.6.4, L(A) is pseudocomplemented. By the above discussion, together with the assumed completeness of A and Lemma 6.6.4(4), for every element z E A, the ideal Jz1. coincides with (zoO]. By Corollary 1.5.6, z" E B(A). 0 With the above notation, a straight forward computation yields

(6.17) x $ xoOoO; (6.18) If x $ y then xoOoO $ y .... ;

(6.19) (x A y)oOoO

= x .... A y**;

(6.20) If z E B(L) then z .... = z. Theorem 6.7.3 Let A be a eomplete MV-algebra and z an atom of B(A). 1f there is an atom a of A sueh that a $ z, then the MV-algebra (z] is isomorphie to the finite ehain Ln, for some n ~ 2. 1f no sueh atom exists, then (z] is isomorphie to [0, 1].

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CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

is a prime ideal in L(A). Let us assume x A y ::; -.z. By (6.17)-(6.20), Proo/: We first prove that

x** Ay**

=

(-.z]

(x Ay)**::; (-.z)**

=

-.z.

Sinee, by Lemma 6.7.2, both x** and y** are elements of B(A), we must either have x** ::; -.z or y** ::; -.z. By (6.17), either x::; -.z or y ::; -.z. Therefore, (-.z] is a prime ideal of L(A) and, by Corollary 1.5.6, it is also a prime ideal of A. Thus the quotient MV-algebra AI (-.z] is totally ordered. By Corollary 6.4.1, AI (-.z] is isomorphie to (z]; by Corollary 6.6.5(ii), (z] is eomplete. Sinee a is an atom of (z] iff ais an atom of A and a::; z, the desired result now follows from the observation that [0,1] is the only simple eomplete atomless MV-algebra. 0

Lemma 6.7.4 For each atom a 0/ a complete MV-algebra A, a** is an atom 0/ the boolean algebra B(A). Proof: By Lemma 6.7.2, a** E B(A) and, by (6.18), a** # O. Let us assume that z E B(A) and z ::; a**. By (6.19) and (6.20), z = z A a** = z** A a**

=

(z A a)**,

and sinee a is an atom of A, we must either have z /\ a = 0 or z /\ a = a. Therefore, z = 0 or z = a**. 0

6.8

Complete distributivity

As usual, a eomplete lattice L is said to be completely distributive iff for every family {Ji I i E I} of nonempty sets, and for arbitrary Xij E L, letting T = [LeI Ji we have

/\ V Xij = V /\ Xif(i) feTieI

and

V /\ Xij = /\ VXif(i) . feTieI

We say that a eomplete MV-algebra A is completely distributive iff its underlying lattice L(A) is eompletely distributive in the above sense.

6.8. COMPLETE DISTRIBUTIVITY

135

Theorem 6.8.1 For every MV-algebra A the following conditions are equivalent: (i) A is a direct product of totally ordered complete MV-algebras; (ii) A is complete and completely distributive; (iii) A is complete and the boolean algebra B(A) is atomic. Proof' (i) => (ii): Trivial. (ii) => (iii): By Corollary 6.6.5, the boolean algebra B(A) is eomplete and eompletely distributive. Then, by a classieal result of Tarski, B(A) is isomorphie to apowerset boolean algebra. Thus in partieular, B(A) is atomie. (iii) => (i): The atoms of B(A) form a set {Zi I i E I} sueh that Zi 1\ Zj = whenever i =J j; further, ViEl Zi = 1. By Lemma 6.6.6, A is isomorphie to the product MV-algebra ITiEI( zd. By Theorem 6.7.3, eaeh (Zi] is a eomplete ehain. 0

°

Corollary 6.8.2 An MV-algebra is the direct product of copies 0/[0,1] iff it is complete, completely distributive, and atomless. 0 Corollary 6.8.3 An MV-algebra is a direct product of finite chains iff it is complete and atomic. 0 As a eorollary, we have a new proof ofthe result (Proposition 3.6.5) that every finite MV-algebra is a direet product of finite ehains. Definition 6.8.4 An MV-algebra A is said to have no linear factors iff whenever A ean be written as a direct product of MV-algebras,

then no Ai is totally ordered. For any eomplete MV-algebra A let {ai I i E I} be the set of atoms of A. Let {Zj I j E J} be the set of atoms of B (A). Then we define SA

=

Va** i

iEI

,

and

ZA

=

V

Zi'

iEI

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

136

In the rest of this section, whenever in a product MV-algebra one of the factors (xl coincides with the singleton (0], we shall tacitly understand that (xl must be deleted.

Theorem 6.8.5 I/Ais a complete MV-algebra then

I/ SA > 0 then (SA] is a direct product 0/ finite chains. I/ SA i- ZA then (-'SA A ZA 1 is a direct product 0/ copies 0/ [0, 1], and the MV-algebra (-'ZA 1 is complete and has no linear /actors. Proof" Formula (6.21) immediately follows from Lemmas 6.6.6 and 6.7.4. If 0 i- Z E B( -'ZA], then Z is not an atom of B(A) and, by Proposition 6.4.3, (z] is not totally ordered. By Lemma 6.4.4, (-'ZA] has no linear factors. Assume SA =1= 0 and let x E (SA], x i- O. If ai i x for each i E I we obtain x

= x A S A = x A Va;* iEI

iEI

iEI

which is impossible. Therefore, ai ~ x, for some i E I, whence (SA 1 is a complete atomic MV-algebra. By Corollary 6.8.3, (SA] is a direct product of finite chains. By Lemma 6.7.4, SA ~ ZA· Thus if SA i- ZA, then -'SA A ZA i- O. If a were an atom of (-'SA A ZA ], then a would also be an atom of A; then a ~ a** < SA, whence a = 0, which is impossible. Therefore, (-'SA A ZA ] is an atomless complete MV-algebra. The set of atoms of B((-'SA A ZA ]) = (-'SA A ZA] n B(A) coincides with the set of those atoms Zi of B(A) such that Zi ~ -'SA. From the identities

-'SA

A

ZA

=

-'SA

A

VZi =

iEI

V(-'SA

A

Zi)

iEI

it follows that B(( -,sAAzA ]) is an atomic boolean algebra. By Theorem 6.8.1 and Corollary 6.8.2, if SA =1= ZA then (-,sAAzA] is a direct product of copies of [0,1]. 0

6.9. BIBLIOGRAPHICAL REMARKS

6.9

137

Bibliographical re marks

In every MV-algebra A the operation -, satisfies the following conditions: (MI) -,0

= 1;

(M2) -,-,x

= x;

(M3) -,(x V y) = -,x A -'y. A distributive lattice with 0 and 1 equipped with an operation satisfying (MI), (M2) and (M3), is called De Morgan algebra. Ais called a Kleene algebra (see [12)) Hf it satisfies the additional condition:

By the subdirect representation theorem, for every MV-algebra A, its underlying lattice (A, V, A, -', 0,1) is a Kleene algebra. Theorem 6.1.3 was first proved in [146], and was used in [99] and [100] to show that, in Zermelo-Fraenkel set theory without the axiom of choice, the following statements are equivalent (one implication is trivial): • Every MV-algebra has a maximal ideal. • Every boolean algebra has a maximal ideal. It is known that this latter statement is strictly weaker than the axiom of choice. The result proved in the claim in Theorem 6.1.5 is a well known elementary fact of the theory of distributive lattices, due to Birkhoff and Stone. See, e.g., [103]. For a study of stonean ideals in lattices see [45]. The proof of the claim in Proposition 6.3.3 can be found in [163, Lemma 8.5]. Proposition 6.4.1 is due to Rodriguez, see [212]. For direct product decompositions also see [123]. For boolean products of MV-algebras see [231], [232], [55] and [56J. Corollary 6.5.6 is due to Torrens [231J (caution: our hyperarchimedean MV-algebras are called archimedean in [231]).

138

CHAPTER 6. LATTICE-THEORETICAL PROPERTIES

Proposition 6.6.2 is due to [132]. In the same paper the author also proved Lemma 6.6.4. Also see [49]. The direct algebraic proofpresented here is due to [14]. Equations (6.17)-(6.20) hold in every distributive pseudocomplemented lattice (see [103], or [12]). For a proof of the c1assical result of Tarski referred to in the proof of Theorem 6.8.1 see, for instance, [25], Chapter 5, §5, Theorem 17. Corollary 6.8.2 is due to Bosbach (see [31]). In the same paper one can also find a proof of Corollary 6.8.3 (see also [49]). Complete MV-algebras, convergence properties, various kinds of topological and order completions are considered, e.g., in [91], [89], [124], [14], [15], [128], [125], [126], [69], [224], [119] and [127]. For model-completions see [135] and [136].

Chapter 7 MV-algebras and f-groups As proved at the beginning of Chapter 2, r is a functor from the category A of f-groups with a distinguished strong unit, to the category MV of MV-algebras. In this chapter we shall prove that r is a natural equivalence (i.e., a full, faithful and dense functor) between A and MV. As a consequence, a genuine addition can be uniquely recovered from the MV-algebraic structure. Several applications will be discussed.

7.1

Inverting the functor

r

In this section we shall give an explicit construction of an adjoint functor of r. Our starting point is the f-group GA with order unit UA considered in Section 2 of Chapter 2. As the reader will recall, for every MV-algebra A, GA is an f-group and there is an isomorphism between the ordered monoid GA + of its positive elements and the ordered monoid M A of good sequences of A. Let A and B be MV-algebras, and h: A ~ B a homomorphism. If a = (al, a2, ... ) is a good sequence of A, then (h(ad, h(a2), ... ) is a good sequence of B. Let h*: M A ~ MB be defined by

(7.1) h*(a) = (h(al), h(a2), .. .), for all a E M A . By direct inspection, using (2.11) and Proposition 2.2.3, for all a, b E M A we have

(7.2) h*(a + b) = h*(a)

+ h*(b)j 139

R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

140

CHAPTER 7. MV-ALGEBRAS AND l-GROUPS

(7.3) h*(a V b) = h*(a)

V

h*(b);

(7.4) h*(a/\ b) = h*(a) /\ h*(b). Thus, h* ; M A -+ MB is both a monoid-homomorphism and a latticehomomorphism. Let us further define the map h# ; GA -+ G B by

(7.5) h#([a, b]) = [h*(a), h*(b)]. Let UA and UB be the strong units of GA and G B given by Proposition 2.4.4. Then by (7.2)-(7.4), the map h# is a unitall-group homomorphism of (GA, UA) into (G B, UB). Let us agree to write

(7.6) E(A) = (GA,UA) and E(h) = h#. Proposition 7.1.1 E is a functor from MV into A. 0 In our present notation, Theorem 2.4.5 states that the map a.- 'PA(a)

= [(a), (0)] defines an isomorphism ofthe MV-algebras A and r(E(A)).

Using the maps 'PA (A E MV), we obtain

Theorem 7.1.2 The composite functor rs is naturally equivalent to the identity functor of MV. In other words, for all MV-algebras A, B and homomorphism h ; A -+ B, we have a commutative diagram h

A

--+

'PA ! r(E(A))

r(S(h»

--+

B

! 'PB r(E(B))

in the sense that, for each a E A, 'PB(h(a)) = (f(E(h)))('PA(a)). Proof: For each a E A, 'PB(h(a)) = [(h(a)), (0)] and 'PA(a) = [(a), (0)]. Further, by (7.1)-(7.6) E(h)([(a), (0)]) = [(h(a)), (0)], the latter being an element of r(S(B)). Since r(E(h)) is the restrietion of S(h) to r(E(B)), we can write r(E(h))('PA(a)) = [(h(a)), (0)] = 'PB(h(a)), as required. 0 Our next aim is to prove that the composite functor sr is also naturally equivalent to the identity functor of the category A. We first prepare the "dual" of the maps 'PA.

7.1. INVERTING THE FUNCTOR

r

141

Lemma 7.1.3 Suppose Gis an f-group with order unit u, and let A = r(G, u) ~ G. For each 0 :s; a E G there is a unique good sequence g(a) = (al,"" an) 0/ elements 0/ A such that a = al + ... + an.

Proo/: For every integer k 2: 1, let ak be inductively defined by: al = a 1\ u, and ak+1 = (a - al - ... - ak) 1\ u. In order to prove that (al,"" ak,"') is a good sequence of A, we first prove the identity

(7.7) a - al - ... - ak

= (a -

ku)

V0

The proof is by induction on k. For k

= (a = 1,

ku)+.

we have

a - al = a - (a 1\ u) = 0 V (a - u) = (a - u)+. For the induction step, assume the identity to be true for k

= m.

Then

= (a - mu)+ - ((a - mu)+ 1\ u) = 0 V ((a - mu)+ - u) = (a - (m + l)u)+, which settles (7.7). Since for each k 2: 1, ak = (a - (k - l)u)+ 1\ u, then necessarily each ak is in the unit interval [0, u] of G. Since u is a strong unit of G, there is an integer n 2: 1 such that a :s; nu, and hence, aj = 0 for all j > n. We shall now prove that ak EB ak+l = ak. To simplify the notation, let t = a - ku. Then by (7.7)

= ((t+ 1\ u) + ((t - u)+ 1\ u)) 1\ u = (t+ + (t - u)+) 1\ u = ((t V 0) + ((t - u) V 0)) 1\ u

CHAPTER 7. MV-ALGEBRAS AND f-GROUPS

142

= ((2t - u) /\ u) V (t+ 1\ u) = t+ 1\ u = ak. We have used the inequality t+ 2:: ((2t - u) 1\ u), which follows from

(t V 0) - (( 2t - u) 1\ u)

= (u - t) V (t - u) V (u - 2t) V -u 2:: (u - t)

V

(t - u) =

It - ul 2:: o.

Hence (al,"" an) is a good sequence of elements of A n

a - Lai

= an+l =

an +2

= ... =

= [0, u], and

O.

i=l

Finally, to prove uniqueness, note that if (bI, ... ,bm ) is a good sequence of elements of [0, u] such that a = :E~l bi , then by definition of good sequences together with Lemma 2.1.3(ii), we have

and, proceeding inductively, for i 2:: 2,

= (a - bl

-

... -

bi-d 1\ u

as required. 0 It follows from the above lemma that the correspondence a ~ g(a) defines an injective mapping from the positive cone G+ of G, onto the monoid Mr(G,u) of good sequences of r(G, u). In order to show that this mapping is both a monoid-isomorphism and a lattice-isomorphism, we prepare the following

7.1. INVERTING THE FUNCTOR

r

143

Definition 7.1.4 An f.-ideal of an f.-group G is a subgroup J of G satisfying the following condition: (7.8) If

xE J

and

lyl ~ lxi

then

y E J.

Letting for each a E G, al J = a + J be the coset determined by a, and defining al J V bl J = (a V b)1 J, the quotient group GI J becomes an f.-group. The map qJ: G ~ GI J sending each a E G to the coset al J, is an f-homomorphism such that K er(qJ) = J. Moreover, if u is a strong unit of G, then qJ(u) is a strong unit of GIJ. Conversely, if G and H are f.-groups and f: G ~ H is an f.-homomorphism, then Ker(f) = f- 1 ({O}) is an f.-ideal of G, and GIKer(f) is isomorphie to the f-subgroup f(G) of H. An f-ideal J of an f-group G is said to be prime Hf J is proper (Le., J =f G), and the quotient f.-group GI J is totally ordered. Lemma 7.1.5 Let G be an f.-group with an order unit u. Then the mapping g: G+ ~ MrCG,'I.I.) of Lemma 7.1.3 satisfies the following conditions, for all a, b E G+:

(i) g(a + b) = g(a)

+ g(b);

(ii) g(a Vb) = g(a)

V g(b);

(iii) g(a /\ b)

= g(a) /\ g(b);

(iv) g(u) = (u). Proof: The notion of a subdirect product of f.-groups is, mutatis mutandis, the same as the corresponding not ion for MV-algebras (Definition 1.3.1). The following claim is a classieal result of Birkhoff : Claim. Every f.-group G is a subdirect product of totally ordered abelian groups. Indeed, let 0 =f a E G. An adaptation of the proof of Proposition 1.2.13 shows that any f.-ideal of G whieh is maximal for not containing a is automatically prime. It follows that the intersection of all prime f.-ideals of Gis the zero ideal, and G can be embedded into the product

CHAPTER 7. MV-ALGEBRAS AND f-GROUPS

144

of all its prime quotients G / J. Thus, G is a subdirect product of totally ordered f-groups, and our claim is proved. Let X be the set of prime f-ideals of G. Our claim yields a family {GX}XEX of totally ordered abelian groups and a one-one f-homomorphism h:G ~ TIxEXGx such that, for each x E X, the composite function 7rx h maps G onto Gx (7r x being the canonical projection onto G x ). For every a E G, letting ax = 7rx (h(a)), it follows that U x is a strong unit of G x. Given arbitrary elements a, b E G+, let us write g(a) = (al,"" am ) and g(b) = (bI,"" bn ). Then, for each x E X, both (alx,"" amx ) and (blx,"" bnx ) are good sequences in r(G x , u x ). Moreover, n

and

bx

= L:bix ' i=I

Since the MV-algebra r(G x , u x ) is totally ordered, by Proposition 2.2.1 every good sequence of A has the form (l P , a). Thus, there are elements O'x, ßx E Gx with 0 ~ O'x < U x and 0 ~ ßx < u x , satisfying the equations (aIX, ... ,amx ) = (u~"',O'x) and (blx, ... ,bnx ) = (u~"',ßx). Further, by definition of g,

whence (a+b)x = ax+bx = (Px+qx)ux+O'x+ßx. Recalling (2.12), by an application of Lemma 2.1.3(i), for all x EX we obtain (u~"',O'x)

+ (ui"',ßx) = (u~",+q"',O'x$ßx,O'x 0ßx).

Then, by direct inspection, (a + b)x = (aIX"'" amx ) + (b lx , ... , bnx ). We conclude that g(a + b) = g(a) + g(b), and (i) is proved. To prove (ii), for any x E X, we either have ax ~ bx or bx ~ a x . Suppose, without loss of generality, ax ~ bx . Then, from 0 ~ O'x ~ ßx < u x, by (7.9) we get Px ~ qx, whence (u~"', O'x) ~ (u~"', ßx). Therefore, letting r = max(m, n), r

(a V b)x

= :~:)aiX V bix ). i=l

7.1. INVERTING THE FUNCTOR

r

145

In other words, a V b = Ei=l (ai V bi ), whence, by Lemma 7.1.3, g(a V b) = g(a) V g(b), which completes the proof of (ii). The proof of (iii) is similar, and (iv) follows at once by definition of g. 0 Corollary 7.1.6 For every (G, u) E A let the map 'IjJ(G,u) : G GrCG,u) be defined by

-+

for all a E G. It follows that 'IjJ(G,u) is an f.-group isomorphism of G onto GrcG,u), and 'IjJ(G,u)(u) = [(u), (0)]. 0 In the light of Corollary 7.1.6, using the maps '1jJ(G,u) (for all (G, u) E A), we have the following Theorem 7.1.7 The composite functor sr is naturally equivalent to the identity functor of the category A. In other words, for any two fgroups with strong unit (G, u) and (H, v) and uni tal f-homomorphism f: (G, u) -+ (H, v), we have a commutative diagram

(H,v) ! 'IjJ(H,v) S(r(H, v))

(G,u)

'1jJ(G,u) ! S(r(G,u))

in the sense that, for all a E G, 'IjJ(H,v)(j(a)) = S(r(j))('IjJ(G,u)(a)). Proof" By Lemma 7.1.3 we can write g(a+) = (al, ... , an), for a uniquely determined good sequence (al' ... ' an) E MrCG,u). Letting h = r(j), we then obtain f(a)+ = f(a+) =

n

L

i=l

f(ai) =

n

L h(ai), i=l

whence, recalling (7.1), g(j(a)+) = (h(al)' ... ' h(an)) = h*(g(a+)}. Similarly, g(j(a)-) = h*(g(a-)), whence by (7.5), (7.6) and (7.10),

'IjJ(H,v) (j(a))

146

CHAPTER 7. MV-ALGEBRAS AND f-GROUPS = [g(f(a)+), g(f(a)-)] = [h*(g(a+», h*(g(a-»]

= h#([g(a+), g(a-m = 8(r(f» ([g(a+), g(a-m = 8(r(f» ('I/l
7.2

Applications

Lemma 7.2.1 Let h: r(G, u) -+ reH, v) be a homomorphism 01 MValgebras. (i) There is a unique unital f-homomorphism I: (G, u) -+ (H, v) such that h = r(f); (ii) 11 h maps r(G,u) onto r(H, v) then 1 maps G onto Hj (iii) 11 h maps r(G, u) one-one into reH, v) then 1 maps G one-one into H. Prool: (i) In the light of Corollary 7.1.6 and Theorem 7.1.7, let defined by

1 be

1 = 'I/lw,v) (8(h»'I/l
I(a) = 'I/lw,v) ([h*(g(a+», h*(g(a-»]). In the partieular esse when a E [0, u), sinee g(a+) [h*(g(a+», h*(g(a-))] = [(h(a», (0)], whenee

I(a) = 'I/lw,v) ([(h(a», (0)]) = h(a).

= g(a) = (a) then

7.2. APPLICATIONS

147

Therefore, r(f) = h. In order to establish the uniqueness of I, suppose I(b) #- f'(b) for some bE G+, and unital i-homomorphism

1': (G, u)

--+

(H, v).

Writing b = al + ... + ak for suitable ai E r(G, u), if r(f) = r(f') we have I(b) = I(al) + ... + I(ak) = I'(ad + ... + f'(ak) = f'(b), a contradiction. (ii) Suppose that h maps r(G, u) onto r(H, v), and let b E r(H, v). Then there exists an element a E r(G, u) such that h(a) = band I(a) = h(a) = b. Since every element of H+ (resp., every element of G+) is a sum of elements of the unit interval [0, v] (resp., a sum of elements of [0, uD we have that I maps G+ onto H+, whence I maps G onto H. (iii) Finally, if I is not one-one then I maps some nonzero element x E G into the zero element of H. Without loss of generality, ~ x, whence = I(x /\ u) = h(x /\ u), and h is not one-one. 0

°

°

Let us agree to denote by T(G) the set of all i-ideals of G, ordered by inclusion. Theorem 7.2.2 Let G be an i-group with strong unit u, and A r (G, u). Then the correspondence

cp : J

~

cp( J) = {x E G

I Ixl/\ u E

=

J}

is an order-isomorphism /rom the set T(A) 01 ideals 01 the MV-algebra A, ordered by inclusion, onto T( G). The inverse isomorphism 'IjJ is given by H E T(G) ~ 'IjJ(H) = H n [0, u]. Proof" Trivially, 'IjJ(H) E T(A) whenever H E T(G). Conversely, to prove that for each ideal J of A, cp(J) is an i-ideal of G, assume x, y E cp(J). Then (Ixl/\ u) ffi (lyl/\ u) E J. Furthermore,

(Ixl/\ u) ffi (lyl/\ u)

= ((Ixl/\ u) + (lyl/\ u)) /\ u = ((Ixl/\ u) + lyJ) /\ ((Ixl/\ u) + u) /\ u

148

CHAPTER 7. MV-ALGEBRAS AND i-GROUPS =

((lxi /\ u) + Iyl) /\ u

= (lxi + Iyl) /\ u ~ Ix - yl /\u, whenee Ix - yl /\ u E J, and henee, x -

°

y E 4>(J). Sinee E 4>(J), then 4>(J) is a subgroup of G, and sinee 4>(J) satisfies (7.8), 4>(J) is an i-ideal of G. Trivially, the funetion 4>: I(A) --+ I(G) preserves inc1usions. Now it is not hard to see that 1/J(4)(J)) = J, for each ideal J of A. Henee to eomplete the proof, there remains to be shown that for each i-ideal H of G, 4>(1/J(H)) = H; stated otherwise, H = {x E G Ilxl /\ u EH n [0, The inc1usion H S; {x E G Ilxl /\ u E H n [0, is c1ear from (7.8). On the other hand, if lxi /\ u E H n [0, u], then sinee u is a strong unit there is an integer n ~ 1 such that lxi ~ nu. Thus, ~ lxi = lxi /\ nu ~ n(lxl /\u) E H, and x E H. 0

un

un.

°

Corollary 7.2.3 The map J 1--+ J n [0, u] defines an isomorphism between the partially ordered set 0/ prime i-ideals 0/ G and the partially ordered set 0/ prime ideals 0/ r( G, u), both sets being equipped with the inclusion ordering. 0 Theorem 7.2.4 Let G be an i-group with strong unit u. Then /or every i-ideal J 0/ G, we have the isomorphism

r(GIJ,uIJ) ~r(G,u)/(Jn [O,u]). Proo/: Let /: G -+ GI J be the natural i-homomorphism, and k = r(f). Let A = r(G, u) and 1= Jn[O, u]. By Theorem 7.2.2, I is an ideal of A. Sinee by Lemma 7.2.1(ii), k is a homomorphism of A onto r(GI J, ul J), whose kernel eoincides with I, then AI I ~ r (GI J, u I J), as required.

o

Remark: It follows at onee that the eorrespondenee J 1--+ J n [0,1] defines a 1-1 mapping from the set of maximal i-ideals of G onto the set of maximal ideals of r(G, u). We shall noweonstruet uneountably many non-isomorphie simple MV-algebras: We first prepare the following well-known result; its proof is inc1uded here to inerease readability.

7.2. APPLICATIONS

149

Proposition 7.2.5 Let G and H be f-subgroups of the additive group R of real numbers with natural order. Assume 1 E G n H. Then there is at most one f-isomorphism f ofG onto H such that f(l) = 1. Whenever such f exists, then G necessarily coincides with H, and f is the identity function on G. Proof: By way of contradiction, assume G ~ H, and let a E G\H. Then a =1= f (a) EH. In case a < f (a) there is a rational number p/q such that a < p/q < f(a), Le., qa < p < qf(a). Therefore, o < p - qa E G, and 0 > p - qf(a) = pf(l) - qf(a) = f(p - qa) E H, whence f does not preserve the order, a contradiction. In a similar way, if f(a) < a, a negative element of G would be mapped by f into a positive element of H. We have proved that G ~ H. Symmetrically, H ~ G, and we are done. 0 Recalling now Lemma 7.2.1 we obtain

Corollary 7.2.6 Two subalgebras A and B of [0,1] are isomorphie iff A = B; the identity function is the only automorphism of A. 0 Example: As in the example following Theorem 3.4.9, for any irrational number E [0,1] let the MV-algebra Sa be defined by

°

Sa = {m+na I m,n E Z,O ~ m+na ~ I}.

Then it is not hard to see that, for any irrational 0 < ß < 1, Sa = Sß if and only if = ß or = 1 - ß. As a matter of fact, assume Sa = Sß and write 0 < 0 < ß < 1 without loss of generality. By definition, = m + nß and ß = p + qa. It follows that = m + np + nqa, whence m + np = 0 and nq = 1. In case n = q = 1, we get the contradiction 0 < ß - a = p < 1. In the remaining case when n = q = -1, we get a = m - ß < 1, whence m = 1, as required. By Corollary 7.2.6, we then have

°

°

°

°

Corollary 7.2.7 There are uncountably many nonisomorphie simple subalgebras of [0, 1] with one generator. 0

150

CHAPTER 7. MV-ALGEBRAS AND f-GROUPS

7.3

The radical

Let G be an f-group. Given elements a, b in G+, we say that a is injinitely smaller than b, in symbols, a « b, iff na :5 b for each integer n~O.

Lemma 7.3.1 For any f-group G with strong unit u, we have: (7.11) Rad(r(G, u» = {x E G+ I x« u}. Proo/: By Proposition 3.6.4, for every MV-algebra A we have Rad(A) = Injinit(A) U {O}. Let H = {x E G+ Ix «u}. If x E H then a fortiori, x E [0, u]. For each integer 0 :5 n we have (n+ l)lxl :5 u; using the notational convention of (2.7), we can write

n.lxl = u 1\ nix I :5 u 1\ (u -lxI) = -,x, whence H ~ Injinit(r(G,u». Conversely, let x E r(G, u) be infinitesimal. By induction on n we shall prove that nx :5 u. The case n = 0 is trivial. For the induction step, suppose nx :5 u. Then nx = u 1\ nx = n.x :5 u - x, whence (n + l)x :5 u. In conclusion, x « u, as required to complete the proof.

o

Recall from Theorem 7.2.2 the definition of the map 4>.

Lemma 7.3.2 Let J be an ideal 0/ the MV-algebra r(G, u) such that J ~ Rad(r(G, u». Then 4>(J) = {x E G Ilxl E J}. Proof: In the light of Proposition 3.6.4, we must prove that whenever Ixll\u is infinitesimal then lxi :5 u. As a matter offact, by Lemma 7.3.1, if Ixll\u is infinitesimal, then Ixll\u« u. Since u is a strong unit of G, there is 0:5 mEZ such that lxi :5 mu. Therefore, lxi :5 m(lxll\u) :5 u.

o

Let A be an MV-algebra and J an ideal of A contained in Rad(A). With reference to Theorem 2.4.5, since 'PA: A -+ r(G A, UA) is an isomorphism, it follows that 'PA(J) is an ideal of r(G A, UA) contained in Rad(r(GA, UA». As an f-ideal of GA, 4>('PA(J» is an f-group (more precisely, an f-subgroup of GA)' Since by Lemma 7.3.2, 4>('PA(J» ~ {x E GA Ilxl « UA} ~ [0, UA], we conc1ude that 'PA(J) = (4)('PA(J)))+. As a consequence we have

7.4. PERFECT MV-ALGEBRAS

Lemma 1.3.3 Let A be an MV-algebra and J an ideal J ~ Rad(A). Then we have

151

0/ A such that

(i) Por all x, Y E J,
Proo/: Suppose x, Y E J. Since
o

Theorem 1.3.4 Let A be an MV-algebra. Por each ideal J ~ Rad(A) there is an l-group G(J), unique up to isomorphism, such that (J, ffi, 0) is isomorphie to the ordered monoid G( J)+ 0/ positive elements 0/ G(J).

Proo/: Put G(J) =def 4>(
7.4

Perfeet MV-algebras

Let A be an MV-algebra. As an intersection of (maximal) ideals of A, Rad(A) is itself an ideal of A. Let the f-group G(Rad(A» be as in Theorem 7.3.4. If A is semisimple, then G(Rad(A» = {O}. At the other extreme, one can naturally investigate MV-algebras having as many infinitesimals as possible. This leads to the following

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CHAPTER 7. MV-ALGEBRAS AND f-GROUPS

Definition 7.4.1 An MV-algebra A is called perfeet iff A is nontrivial and A = Rad(A) U -,Rad(A), where -,Rad(A) = {x E A I-,x E Rad(A)}. For each perfect MV-algebra A, let ~(A) =def G(Rad(A)),

and for each homomorphism h from A into a perfect MV-algebra B, let ~(h) =def 8(h)I{XEGA Ilxl«UA}

be the restriction of S(h) to the set {x E GA Ilxl

«

UA}.

Since f-group homomorphisms preserve the « relation, it follows that ~(h) is an f-group homomorphism from ~(A) into ~(B). It is easy to check that ~ is a functor from the the category of perfect MV-algebras (considered as a full subcategory of the category of MValgebras) into the category of f-groups. Conversely, we shall give a method for constructing perfect MV-algebras from f-groups. To this purpose, let H be a totally ordered abelian group and G be an f-group. The lexicographic product Ho G of Hand G is the f-group whose elements are the pairs (x, a) such that x E Hand a E G, with the group operations defined pointwise and the order relation given by (x, a) < (y, b) iff either x < y or x = y and a < b. Note that the infimum of two elements in H 0 G is obtained as follows:

(7.12) (x, a)

1\

(x,a) (y, b) = { (y, b) (x,al\b)

if if if

x< y y

x

<x =

y

For each t'-group G, we write A( G) = ZoG, where, as usual, Z denotes the additive group of integers with natural order. Further, for any f-group Hand f-group homomorphism h: G ~ H, let us define the function A(h): A(G) ~ A(H) by

(7.13) A(h)((m, a)) = (m, h(a)) for each pair (m, a) E ZoG. Then A(h) is an f-group homomorphism. As a matter of fact, Ais a functor from the category of lattice-ordered abelian groups into itself.

153

7.4. PERFECT MV-ALGEBRAS

Let G be an f-group. Since for each m > 0 and each a E G, the pair (m, a) is a strong unit of ZoG, we can write ~(G) =

r(A(G), (1,0)).

The zero element of ~(G) is (0,0) and the operations E9 and -, are defined as folIows:

(O,a+b), (i,a)E9(j,b)= { (l,(a+b)/\O), (1,0),

if if if

i+j=O i+j=l i +j = 2

and

-,(i, a)

= (1 - i, -a).

By definition of lexicographic order it follows that ~(G)

= {(O, a) la E G+} U {(1, b) I bE G-}.

The MV-algebra C = ~(Z) is a first example of a nonsemisimple MV-algebra. More generally, since Rad(~(G)) = {(O, a) la E G}, and for each a E G, (1, -a) = (1,0) - (0, a) = -,(0, a), the algebras ~(G) are examples of perfeet MV-algebras. Lemma 7.4.2 Por each perfeet MV-algebra A, the stipulation

if xE Rad(A) if x E -,Rad(A) defines an isomorphism ClA from A onto

~(ß(A)).

Proof' Since
154

CHAPTER 7. MV-ALGEBRAS AND f-GROUPS

Gase 1: x EB Y E Rad(A). Then both x and Y are in Rad(A), and equation (7.14) follows at once from Lemma 7.3.3(i). Gase 2: x EB Y E -,Rad(A). Subcase 2a: x E -,Rad(A) and Y E -,Rad(A). Then by Lemma 7.3.3(ii), aA(xEBy) = (1,-
If h is an f-group homomorphism from G into an f-group H, it follows that A(h): A(G) -+ A(H) is an f-group homomorphism such that A(h)((l,O)) = (1,0). Therefore the map E(h) defined by E(h) =def r(A(h)): E(G) -+ E(H) is a homomorphism. It is now easy to see that E is a functor from the category of f-groups to the category of MV-algebras.

For each f-group G, we have the identity Rad(E(G)) = {O} x G+. It follows that

((E(G))) = {(m, x)

E

A(G)

II(m, x)1 «

(1, O)}

= {O} x G,

whence the mapping a 1---+ ßG(a) = (0, a) is an f-group isomorphism from G onto ß(E(G)). The next lemma shows that the composite functor ßE is naturally equivalent to the identity functor of the category of f-groups. Lemma 7.4.3 For all f-groups G, Hand for any f-group homomorphism h: G -+ H, we have a commutative diagram

G ßG

1

~

ß(E(G)) ~~»

H

1 ßH

ß(E(H))

7.4. PERFECT MV-ALGEBRAS

in the sense that, for each a E G, ßH(h(a)) =

155 .6.(~(h))(ßA(a)).

Proof: It is sufficient to observe that ßH(h(a)) = (0, h(a)), and .6.(~(h) )(ßa( a))

= .6.(~(h))(O, a) = (0, h(a)). o Conversely, the following lemma shows that the composite functor is naturally equivalent to the identity functor of the category of MV-algebras: ~.6.

Lemma 7.4.4 For all MV-algebras A, Band for any homomorphism

h: A

-+

B, we have a commutative diagram

in the sense that, for each a E A, aB(h(a)) =

~(.6.(h))(aA(a)).

Proof: If a E Rad(A) then, trivially, h(a) E Rad(B) and aB(h(a)) = (0, 'PB(h(a))). On the other hand, ~(.6.(h))(aA(a))

= ~(.6.(h)(O, 'PA(a))) = (0, r(B(h))('PA(a))),

and the result follows from Theorem 7.1.2. 0 From Lemmas 7.4.3 and 7.4.4 we immediately get: Theorem 7.4.5 The.6. functor establishes a natural equivalence from

the category of perfeet MV-algebras, considered as a full subcategory of the category of MV-algebras, and the category of f-groups. 0

156

7.5

CHAPTER 7. MV-ALGEBRAS AND f-GROUPS

Bibliographical remarks

The natural equivalence between MV-algebras and f-groups with strong unit was first established in [163], building on previous work by Chang [38] for the totally ordered case (see also [131]). The proof of Theorem 7.2.4 can be found in [56]. The results established in the remark following Lemma 7.3.3 are due to Belluce [14]. As a generalization of the MV-algebra algebra C originally introduced by Chang in [36], perfeet MV-algebras were introduced in [20]. The natural equivalence between the categories of perfeet MV-algebras and f-groups was established by Di Nola and Lettieri [65] (see also [60]).

Chapter 8 Varieties of MV-algebras A dass C of MV-algebras is said to be a variety (or, an equational class) , iff there is a set E of MV-equations such that for every MV-algebra A, A E C iff A satisfies all equations in E. For instance, when E = 0, we obtain the variety MVofMV-algebras. When E = {x = y}, we obtain the variety of trivial MV-algebras. The main aim of this chapter is to describe all varieties of MV-algebras.

8.1

Basic definitions

A variety C of MV-algebras is called proper iff it is nontrivial and is different from MV. By Corollary 1.5.5, the dass of boolean algebras is the proper equational dass ofMV-algebras determined by E = {x = xEBx}. By Lemma 1.4.5 every variety C is dosed under subalgebras, homomorphic images and direct products. Using this observation we obtain that the following three dasses of MV algebras are not varieties: semisimple, hyperarchimedean, and complete MV algebras. Indeed, since by Corollary 3.6.2 all free MV-algebras are semisimple, from Proposition 3.1.5 it follows that the dass of semisimple MV-algebras is not dosed under quotients. As noted after Definition 6.3.1, the dass of hyperarchimedean MV algebras is not dosed under direct products. Finally, the dass of complete MV algebras is not dosed under subalgebras, because Q n [0, 1] is an incomplete subalgebra of the complete MV algebra

[0,1].

157 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

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CHAPTER 8. VARIETIES OF MV-ALGEBRAS

Let K. be a set of MV-algebras and E be the set of all MV-equations which are satisfied by all MV-algebras in K.. The variety determined by E is said to be generated by K., and is denoted by C(K.). When the set K. eonsists of one MV-algebra A, we shall write C(A) instead of C({A }), and we will eall C(A) the variety genera ted by A. With this notation, the Completeness Theorem 2.5.3 amounts to the identity MV = C([O, 1]). More generally we have

Proposition 8.1.1 1/ A is an infinite subalgebra [0,1], then C(A) = MV.

0/ the

MV-algebra

Proo/: Sinee the operations ....,x = 1 - x and x ES y = min(l, x + y) are eontinuous over [0,1], then for each MV-term r(xl, ... , x n ), the function r[O,I): [0, l]n -+ [0,1] is eontinuous. Therefore, sinee by Proposition 3.5.3, Ais dense in [0,1], the equation r(xb .. . ,xn ) = holds in [0, 1] iff it holds in A. The desired result is now a eonsequenee of the Completeness Theorem. o.

°

Proposition 8.1.2 1/2 :5 no < nl < ... is an infinite sequence natural numbers, then C( {L ni li = 0, 1, ... }) = MV.

0/

Proof: For each a E [0, 1] there is a number k such that 1 :5 k :5 n - 1 and la- n~11 < n':l· The desired result now follows from the eontinuity of the term funetions r[O,I): [0, l]n -+ [0,1]. 0 As a eonsequenee of Theorems 1.3.3 and 1.4.6, every variety algebras is generated by the collection 0/ its MV-chains.

0/ MV-

If A is a proper MV-chain, AI Rad(A) is a simple MV-algebra, beeause Rad(A) is the unique maximal ideal of A. By Theorem 3.5.1, AI Rad(A) is isomorphie to a subalgebra of the MV-algebra [0,1].

Definition 8.1.3 Let A be a proper MV-chain. If for some n ;::: 2, AI Rad(A) 9:! Ln, then we say that A is of rank n. If AI Rad(A) is isomorphie to an infinite subalgebra of [0,1], we say that A is of infinite rank.

8.1. BASIC DEFINITIONS

159

As an example, for each integer n;::: 2 the MV-algebra Ln is a simple MV-chain of rank n. For every totally ordered abelian group G, the algebra :E (G) = r(A(G), (1,0)) introduced in Section 7.4 is a nonsimple MV-chain of rank 2. More generally, for each bEG and each integer n ;::: 2, consider the MV-algebra r(A(G), (n - 1, b)). By definition oflexicographic product, r(A(G), (n - 1, b)) coincides with the union AUBUC, where A is the set of all pairs of the form (0, x), for arbitrary b ~ x E G, B is the set of all pairs (n - 1, y), for arbitrary b ;::: y E G, and C = U~":i2( {i} x G). The pair (0,0) is the zero element, and the operations ..., and E9 given by

(8.1) (i,x)E9(j,y) = {

(i + j, x

+ y)

(n - 1, b)

if i + j < n - 1 or if i+j=n-landx+y
(8.2) ...,(i, x) = (n-l-i,b-x) It follows that the 0 operation of r(A(G), (n - 1, b)) is given by (i+j-n+l,x+y-b) ifi.+.J>.n-l or If 1, + ) = n - 1 (8.3) (i x) 0 (' ) = { , ), Y and x + y > b. (0,0) otherwise.

A straight forward verification shows that r(A(G), (n - 1, b)) is a nonsimple MV-chain of rank n. Indeed, the projection 7l"1: A(G) - Z, defined by 7l"l((j, t)) = j, for each (j, t) E A(G), is a surjective f-group homomorphism, with 7l"l((n - 1, b)) = n - 1. Hence r(7l"l): r(A(G), (n - 1, b)) - r(Z, n - 1) is a surjective homomorphism. Since Ker(r(7l"l)) = {(O, t) ItE G+}

= Rad(r(A(G), (n -

1, b))),

we have r(A(G), (n - 1, b))j Rad(r(A(G), (n - 1, b)))

~

r(Z, n - 1)

f'oj

Ln.

160

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

For each integer n

~

2, we shall write, in abbreviated form,

Kn

= r(A(Z), (n - 1,0))

Rn

= r(A(Z), (n - 1,1)).

and

Proposition 8.1.4 1/ C is a proper variety 0/ MV-algebras then there is an integer m ~ 2 such that rank(A) :$ m /or alt MV-chains A E C. Proof: Immediate from Propositions 8.1.1 and 8.1.2, upon noting that A E C implies AI Rad(A) E C. 0

In the next sections we shall establish that, conversely, if the ranks of all MV-chains in a variety C are bounded, then C 1= MV. We first consider simple MV-algebras of finite rank, i.e., the algebras Ln. The simplicity assumption shall be removed in a subsequent section.

8.2

Varieties from simple algebras

Following the notation in Section 1 of Chapter 2 (see Equation (2.7)), for x E [0,1], we will write n.x instead of x EB ... EB x (n times), and write nx instead of x + ... + x (n times). Definition 8.2.1 Let n ~ 2 be an integer. By an n-bounded MValgebra we shall mean an algebra satisfying the equation (Eno ) (n - 1).x = n.x The variety of n-bounded MV-algebras will be denoted by Uno

Theorem 8.2.2 Let A be an MV-algebra and n ~ 2 an integer. Then A E Un i/ and only i/ A is a subdirect product 0/ algebras L k , with 2:$ k :$ n.

8.3. MV-CHAINS OF FINITE RANK

161

Proof: Let A E Uno By the remark following Definition 3.6.3 we have Rad(A) = (0). Hence A is semisimple, Le., A is a subdirect product of subalgebras of [0, 1]. By Theorem 1.4.6, all these subalgebras satisfy the equation (n - l).x = n.x. Let S be a sub algebra of [0,1] such that, for each k = 2,3, ... ,n, S =f Lk. Then by Proposition 3.5.3 there is t E S such that < t < l/(n - 1). It follows that (n - l).t = min(l, (n -l)t) < min(l, nt) = n.t. Therefore A is a subdirect product of algebras L k , for suitable integers 2 ~ k ~ n. Conversely, let A be a subdirect product of algebras L k , with 2 ~ k ~ n. For all x E L k , we either have x = or x 2: l/(k - 1). From k -1 ~ n -1 it follows that min (1, (n -l)x) = 1 = min (1, nx). Thus, for each 2 ~ k ~ n the MV-chain L k belongs to Uni by Theorem 1.4.6, A E Uno 0

°

°

Recalling now Theorem 1.4.6 we have

Corollary 8.2.3 For each integer n 2: 2, Un

= C( {L2 , L3 , ••• , Ln}).

0

Corollary 8.2.4 An MV-algebra A belongs to the variety C(L 2 ) iJJ A is a boolean algebra, i.e., iJJ A satisfies the equation xEBx = X. An MValgebra belongs to C(L3 ) iJJ it satisfies the equation x EB x = x EB x EB X.

Proof: The first result easily follows from Theorems 8.2.2 and 1.5.3(iv). For the second result, by the remark following Proposition 3.5.3, L 2 is a subalgebra of L 3 • Thus, an equation is simultaneously satisfied by L 2 and by L3 iff it is satisfied by L3 . Thus, C( {L2 , L3 }) = C(L3 ), and the desired conclusion now follows from Theorem 8.2.2. 0

8.3

MV-chains of finite rank

As we have seen above, for every totally ordered group G, algebras of the form r(A(G), (n - 1, b)) yield examples of MV-chains of rank n. We shall see in this section that these are in fact the most general examples of nonsimple MV-chains of rank n. Firstly we introduce some notations. Let A be a totally ordered MV-algebra. By Theorem 7.3.4 we can associate with Rad(A) the f-group G(A), which is an f-subgroup of the

162

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

.e-group GA. Hence, by the remark following Theorem 2.4.5, G(Rad(A)) is a totally ordered group, denoted p(A). When A is a perfect MVchain, then p(A) = a(A).

Theorem 8.3.1 Let A be a nonsimple MV-chain 01 rank n. Then there is 0 < b E Rad(A) such that A ~ r(A(p(A)), (n - 1, c,oA(b))).

Proof: Let I: A ~ Ln be the composition ofthe natural homomorphism from A onto AI Rad(A) and the isomorphism from AI Rad(A) onto Ln. By Lemma 7.2.1, there is a unique .e-group homomorphism g: GA ~ Z such that g(UA) = n - 1 and r(g) = I. Since 1 is surjective, by the

same Lemma we know that 9 is surjective. Since 1 is not injective (because A is not simple), 9 is not injective, that is, K er(g) = {x E GA Ig(x) = O} =f {O}. By Theorem 7.2.4, we have isomorphisms

r(G A, UA) Ic,oA (Rad(A)) ~

r(Z, (n - 1))

~

r(GAI Ker(g), uAI Ker(g))

~

r(G A, uA)I'!jJ(Ker(g)).

Since every (nontrivial) MV-chain has exactly one maximal ideal, we obtain p(A) = 4>(c,oA(Rad(A))) = 4>('!jJ(Ker(g))) = Ker(g). Therefore, to prove the theorem it is sufficient to prove the following

Claim. There is an .e-group isomorphism h: GA h(UA) = (n -l,v), for some 0 < v E Ker(g).

~

A(Ker(g)) such that

To prove the claim, we first show that there is 0 < z E GA such that g( z) = 1 and (n-l)z < UA. Indeed, since 9 is surjective, there is y E GA such that g(y) = 1. Note that y > 0, for otherwise g(y) ::; g(O) = 0, which is impossible. If (n - l)Y < UA, then we can put z = y. If (n-1)y = UA, then we can put z = y-t, where tE Ker(g), t > 0 (note that t < y, for otherwise g(y) ::; g(t) = 0, a contradiction). Finally, if (n - l)y > UA, then (n - l)y = U + 8, for some 0 < 8 E Ker(g), and we can put z = Y - 28. To complete the proof of the claim it is enough to let h be defined by h(x) = (g(x), x - g(x)z), for each x E GA, and v = UA - (n - l)z. 0

8.3. MV-GRAINS OF FINITE RANK

163

Lemma 8.3.2 Por each integer n 2:: 2 and each nonsimple MV-chain A of order n, Rn E C(A). Proof: By Lemma 8.3.1 we can assurne A ~ r(A(G), b) for some totally ordered group G and 0 < bEG. The mapping h: A(Z) -+ A(G) defined by h((j, m)) = (j, mb), for each (j, m) E Z x Z, is an injective l-group homomorphism and h((n - 1,1)) = (n - 1, b). Therefore r: Rn -+ r(A(G), (n - 1, b)) is an injective homomorphism, whence it follows that Rn E C(A). 0

For all integers k and n, n 2:: 2, let the map h: A(Z) -+ A(Z) be defined by h((j, m)) = (j, (n - l)m - kj). Then h is an injective lgroup homomorphism with h((n - 1, k)) = (n - 1,0). It follows that_

r(h): r(A(Z), (n - 1, k))

-+

Kn

is an embedding, whence

(8.4) for all integers k and n, n 2:: 2, r(A(Z), (n - 1, k)) E C(K n ). This result is considerably strengthened by the following proposition: Proposition 8.3.3 Let G be an l-group and 0< bEG. Then for each integer n 2:: 2, r(A(G), (n - 1, b)) E K n . Proof: We need the following c1assical result from the theory of lgroups: Por any lattice-ordered abelian group G there exists a nonempty set I, an l-subgroup H of the l-group ZI and an l-group homomorphism from H onto G.

We sketch here a proof for the reader's convenience. Since G is a subdirect product of totally ordered abelian groups (see the proof of the Claim in Lemma 7.1.5), one can easily adapt the proof of the Completeness Theorem (Theorem 2.5.3) to show that an l-group equation holds in all l-groups Hf it holds in Z. For each nonempty set I let H be the l-subgroup of the l-group ZI generated by the set P of projection functions '7ri: ZI -+ Z, for all i E I. Arguing as in the proofs

164

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

of Lemma 3.1.2 and Proposition 3.1.4, one immediately shows that H has the following property: Each function from the set P to an f-group G can be uniquely extended to an f-group homomorphism from H to G. The desired conclusion now follows by letting I be an arbitrary set whose cardinality is greater or equal than that of G (compare with Proposition 3.1.5). Returning to the proof of the proposition, suppose that G is an fgroup and 0 < bEG. Let H an f-subgroup of ZI and h: H ~ G be a surjeetive f-group homomorphism. Then there is c E H sueh that h( c) = b; replacing, if neeessary, c by c V 0, we ean safely assume c > O. It follows that A(h): A(H) ~ A(G) is a surjeetive f-group homomorphism and A(h)((n - 1, c)) = (n - 1, b). Henee

r(h): r(A(H), (n - 1, c))

~

r(A(G), (n - 1, b))

is a surjeetive homomorphism. Sinee His an f-subgroup of ZI, eaeh element of xE H is a function x: I ~ Z. The funetion f: A(H) ~ A(Z)I defined by f((m, x))(i) = (m, x(i)), for i E I, x E Hand mEZ, is an injeetive f-group homomorphism. Let J = {i E I I c(i) i= O}. Sinee c > 0, J is nonempty; moreover, i E J iff c(i) > O. Therefore r(A(H), (n - 1, c)) is isomorphie to a sub algebra of the produet algebra IIiEJr(A(Z), (n - 1, c(i))). By (8.4), for eaeh i E J, we get r(A(Z), (n - 1, c(i))) E C(K n ). Therefore, r(A(Z), (n - 1, c)) E C(Kn ). Sinee the MV-algebra r(A(G), (n - 1, b)) is a homomorphic image of r(A(Z), (n - 1, c)), we finally get

r(A(G), (n - 1, b))

E

C(K n ),

as required to eomplete the proof. 0

Lemma 8.3.4 For each integer n 2:: 2, K n E C(H n ). Proof" Suppose that an equation does not hold in K n . By the remark following Definition 1.4.4 we can assume that the equation has the form

8.3. MV-GRAINS OF FINITE RANK

165

Xk) = 0, for some MV-term 7r in the variables Xl! ... , Xk. There are elements Cl, ... ,Ck E Kn such that 7r(XI, ••• ,

(8.5)

7r Kn (CI' ... ' Ck)

~ (0,1).

For each integer m ~ 1, let fm: A(Z) -+ A(Z) be the f-group homomorphism defined by fm((v, w)) = (v, mw) for all pairs (v, w) E Z x Z. For every MV-term a in the variables Xl! ... ,Xk let g(a) denote the total number of symbols..., and EB occurring in a. By induction on g(a), for each k-tuple (bl! ... , bk ) E Kn ~ Rn we shall prove the following inequalities: (8.6) fm(aKn(b l , ... , bk )) - (0, g(a))

(8.7) ~

aHn (fm(b l ), ... ,

fm(b k ))

(8.8) ~ fm(aKn(b l , . .. , bk )) + (0, g(a)). If g(a) = 0, then a = Xi, for some i E {1, 2, ... , k} or a = 0. In this latter case, the result is obvious. In the former case, if bi = (v, w) then

fm(aKn(b l , ... , bk))

= fm(b i ) = (v, mw) = aHn (fm(bd, ... , fm(b k)),

thus establishing (8.6) for g(a) = 0. Proceeding by induction, pick an integer d > 0, and suppose that (8.6) holds for all MV-terms ~ in the variables Xl! .. . , Xk such that g(~) < d. Let a(xI, . .. , Xk) be an MV-term such that g(a) = d. From Section 1.4 we know that precisely one of the following two cases must occur:

Gase 1: There is an MV-term 7(XI, ... , tk) such that a = ""7; Gase 2: There are MV-terms /-l(XI, . .. , Xk) and V(XI, . .. , Xk) such that a=/-lEBv. For notational convenience, and without any essential loss of generality, we shalllimit ourselves to considering MV-terms in one variable. In Case 1, we can write g(7) = g(a) - 1 < d, whence, by induction hypothesis,

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

166

= (n - 1,1) - THn(fm(b)) ::; (n - 1,1) - fm(TKn(b))

+ (0, g(7))

= (n -

+ (0, 1 + g(T))

1,0) - fm (T Kn (b))

= fm (u Kn (b)) + (0, g(T)) and

uHn(fm(b)) ~ (n - 1,1) - fm(TKn(b)) - (0, g(T)) ~

fm(TKn(b)) - (O,g(u)).

From these inequalities one immediately sees that u satisfies (8.6). In Case 2 we have g(u) = g(J.L) + g(lI) + 1. By induction hypothesis,

uHn(fm(b)) = (n - 1,1) A (J.lHn(fm(b)) ::; (n - 1,1)

A

+ lI Hn Um(b)))

+ fm (lIK n(b)) + (0, g(J.l)) + (0, g(lI)))

Um (J.lK n(b))

::; (n - 1, g(J.l) + g(lI) + 1)

A(fm(J.lKn(b)) + fm (lIK n(b))

= ((n -

1,0) A (fm (J.lK n(b))

+ (0, g(J.l) + g(lI) + 1)) + fm (lIK n(b))) + (0, g(J.l) + g(lI) + 1)

= fm(uKn(b)) + (0, g(u)) and

uHn(fm(b) ) ~ (n - 1,1) A (fm(J.lKn(b)) ~

+ fm (lIK n(b)) -

(0, g(J.l)) - (0, g(lI)))

(n - 1,1 - g(J.l) - g(u))

A(fm(J.lKn(b)) + fm (lIK n(b)) - (0, g(J.l)) - (0, g(lI)))

= ((n - 1,1) A (fm (J.lK n(b)) + fm (lIK n(b))) - (0, g(J.l) + g(lI))

8.4. KOMORl'S CLASSIFICATION

167

2: ((n - 1,0) A (fm(J.lKn(b)) + fm (l/K n(b))) - (0, g(J.l) + g(a) + 1)

= /m(aKn(b)) - (O,g(a)). Rence the inequalities (8.6) also hold in this case. From (8.5) and (8.6) we obtain (0, m - g(7r)) ~ fm(7r Kn (b 1 , ••• , bk )) ~

7r Hn (fm(b1 ), .•. ,

If m > g(7r), then

-

(0, g(7r))

/m(bk )).

... , /m(b k )) > (0,1) and Rn does not satisfy the equation 7r(Xll ••• ' Xk) = O. We conclude that K n must satisfy all equations that are satisfied by Rn. 0 7r Hn (fm(bt},

From Theorem 8.3.1, Proposition 8.3.3 and Lemmas 8.3.2 and 8.3.4, we obtain Theorem 8.3.5 For each integer n 2: 2 and each nonsimple MV-chain A 0/ rank n, C(A) = C(K n ). 0

8.4

Komori 's classification

The two formulas (8.1) and (8.3) show that, for each (j, a) E K n = r(A(Z), (n - 1,0)) and integer k 2: 1, (8.9) k.(j, a)

=

{

if kj < n - 1 or kj = n - 1 and a<0 (n - 1,0) otherwise

(kj, ka)

and

(kj - (k - 1)(n - 1), ka) if .k~ > (~- 1)(: - 1) or )k = { kJ - (k 1)(n 1) and (810)(. . J,a a> 0 (0,0) otherwise. Lemma 8.4.1 Let n 2: 2 be an integer and A an MV-chain. Then A has rank ~ n i/ and only if A satisfies the equation

168

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

Proof: Let (j, a) E Km. Whenever 0 ~ j ~ m - 2, from (8.10) it follows that (j, a)m-l E Rad(Km ), (j, a)m = (0,0), and (m -1, b)m-l = (m -1, b)m = (m -1,0). Hence the equation (Fno ) holds in the algebra Km for m = 2, ... , n; further, by Theorem 8.3.5, (Fno ) holds in all MV-chains of rank m ~ n. Since AI Rad(A) E C(A), to complete the proof it is sufficient to show that for each m > n, the MV-algebra L m does not satisfy equation (Fno ). To this purpose, we must prove that there is t E L m such that n.tn - 1 = 1 and 2.tn < 1. Note that for each t E L m , n.tn - 1 = 1 iff n((n - l)t - (n - 2)) ~ 1 and 2.tn < 1 iff 2(nt - (n -1)) < 1. Therefore, for each m > n we must find an integer k such that n -1 k 2n -1 -n- < < 2n . m-l

In case n < m ~ 2n we can take k = m - 2. In case 2n < m, there are integers q and r such that q ~ 1, 0 < r < 2n and m = 2nq + r. Thus, letting k = m - (q + 2) we obtain the desired conclusion. 0 Theorem 8.4.2 For each nontrivial MV-algebra A and integer n the following conditions are equivalent:

~

2,

(i) A satisjies the equation

(ii) A E C( {K 2 , K 3 ,· •. , K n }); (iii) AI Rad(A) E C( {L 2 , L 3 , ... ,Ln})' Proof: The equivalence between (i) and (ii) easily follows from the above Lemma, together with Theorems 1.3.3 and 1.4.6. To obtain the equivalence between (ii) and (iii) note that AI Rad(A) is semisimple and that L 2 , ... ,Ln is the complete list of all simple algebras of rank at most n. 0

Corollary 8.4.3 For each nontrivial MV -algebra A we have:

8.4. KOMORI'S CLASSIFICATION

169

(i) A E C(K 2 ) iff A satisfies the equation

(ii) A E C(K 3 ) iff A satisfies the equation

Proof: (i) is an immediate consequence of Proposition 8.4.2, and (ii) follows from the same proposition and the fact that K 2 is a subalgebra of K3 • 0 Remark: It follows from (i) in the above corollary that alt per/ect MValgebras are in C(K 2 ). From Proposition 8.1.4 we know that each proper variety of MValgebras is generated by a set of MV-chains of bounded rank. From Theorems 8.2.2 and 8.4.2 we conclude that, conversely, each set of MVchains of bounded rank generates a proper variety. By Theorem 8.3.5, we then obtain

Theorem 8.4.4 A class C 0/ MV-algebras is a proper variety iff there are two finite sets land J 0/ integers ~ 2 such that Iu J is nonempty and C = C({LihEI, {KjhEJ).O Notation: For each integer n

~

2 we let

Div(n) be the set of all divisors d

~

1 of n.

Lemma 8.4.5 Let m and n be integers such that n ~ m ~ 3. For each integer p such that 2 ::; p < m - 1, the MV-algebra Km satisfies the equations:

(Enp )

(p.xp-1)n = n.xp

and

(Fnp )

n.xp

= (n + l).xp

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

170

if and only if p does not divide m - 1. Proof: Let y = (j, a) be an arbitrary element of Km. Let us assurne p ~ Div(m-1). If yP-l = 0, then yP = 0, whenee (p.yp-l)n = n.yP and n.yP = (n + l).yp. If, on the other hand, yP-l > 0, then there are two subeases to eonsider: If yP = 0, from (8.10), together with our standing hypothesis that p does not divide m - 1, it follows that jp < (m - 1)(P - 1). Therefore, p(p - l)j - p(p - 2)(m - 1) ~ m - 2. Henee from (8.9), we have p.yP-l ~

= (p(p _ l)j - p(p - 2)(m - l),p(P - l)a)

(m - 2,p(p - l)a),

and again by (8.10), (p.yp-l)n = 0 = n.yp. If yP > 0 then jp > (m - l)(p - 1), whenee p.yp-l = 1 = yP and n.yP = n.(pj - (p - l)(m - l),pa)

= (m - 1,0) = (n + l).yp.

Therefore both equations (Enp ) and (Fnp ) hold in Km. Conversely, let us assurne that n 2: p 2: 2. If y = then yP = 0 and yp-l = ~. Therefore L p +1 does not satisfy equation (En(p_l)), and sinee Lp +1 is isomorphie to the subalgebra {(O, 0), (1,0), ... , (p, O)} ~ Kp+1, we also obtain that equation (En(p_l)) does not hold in Kp+1' On the other hand, from z = (p, -1) E Kp+1 and zP = (p, -p) we get n.zP < (n + l).zp, whenee equation (Fnp ) does not hold in Kp +1 . To eomplete the proof it is sufficient to note that if p E Div( m - 1) then K p +1 is a sub algebra of Km. 0

7'

Theorem 8.4.6 Let land J be sets of integers 2: 2 such that IU J is nonempty; suppose that whenever r, sEI u J and r < s then r - 1 ~ Div( s -1). Let n be the greatest element of I U J. Let A be a nontrivial MV-algebra. Then

A E C({LihEI, {KjhEJ) iff A satisfies the equations

(Fno )

(n.x n- 1)2

= 2.xn ,

8.5. VARIETIES GENERATED BY A FINITE CHAIN

(Enp )

(p.xp-1)n

171

= n.xp,

for each integer p such that 2 ~ p < k and p

~ UrEIUJ

Div(r - 1),

as well as the equations

(Fnq )

k.x q = (k + 1).xq,

for each integer q 2: 2 such that q E U(Div(r -1) \ U Div(s -1)). rEI

SEJ

Proof: Let C be the dass of all MV-algebras satisfying equations (Eno ), (Enp ), for each integer p such that 2 ~ p < k and p ~ UrEIUJ Div(r-l), and (Fnq ), for each integer q 2: 2 such that q E UrEI(Div(r - 1) \ UsEJ Div(s -1)). It follows from Lemmas 8.4.1 and 8.4.5, that Km E C if and only m -1 divides s -1 for some s E J. Let r E land y E L r +1 . Since yq E Lr +1, by Theorem 8.2.2 we have (r - 1).yq = r.yq, whence a fortiori, n.yq = (n + 1).yq. Since for each m, L m is isomorphie to a subalgebra of Km, by Lemma 8.4.5 we can now condude that L m is a member of C if and only if k - 1 divides r - 1 for some r E I U J. Hence by Theorem 8.4.4, C = C({LihEI, {KjhEJ). 0

8.5

Varieties generated by a finite chain

Corollary 8.2.4 gives a simple equational characterization of the varieties of MV-algebras generated by L 2 and L 3 • The following theorem, which is an immediate consequence of Theorems 1.4.6 and 8.2.2 together with Lemma 8.4.5, gives a set of defining equations for each variety generated by a finite simple MV-algebra, in all remaining cases. These equations are simpler than those obtainable by letting I = {n} and J = 0 in Theorem 8.4.6.

Theorem 8.5.1 For every integer n 2: 4 and every MV-algebra A, the following conditions are equivalent:

(i) A satisfies the equations (Eno )

(n - l).x

= n.x

172

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

and

(Enp )

(p.xp-1)n = n.xp.

/or every integer p

(ii) A is a produet (iii) A

E

C(Ln ).

= 2, ... , n -

2 that does not divide n - 1;

0/ subalgebras 0/ Ln;

0

Definition 8.5.2 Algebras in the variety C(L n ) are called n-valued MV-algebras, or, for short, MVn-algebras . Corollary 8.5.3 Let A be an MV-algebra.

(i)

1/ A

is a finite MVn -algebra, then A is isomorphie to the direet produet 0/ a /amily 0/ subalgebras 0/ Ln;

(ii) 1/ A is finite then there is an integer n (iii)

~

2 sueh that A E C(L n );

1/ A

is isomorphie to the direet produet 0/ a finite /amily 0/ subalgebras 0/ Ln then the members 0/ this /amily eoineide with the images 0/ A under all possible homomorphisms 0/ A into Ln .

Proo/: (i) By Proposition 3.6.5, Ais isomorphie to the direct product of a finite family of finite chains. Since any such chain L is a homomorphic image of A, L E C(L n ). By Theorem 8.5.1(ii), L must be a subalgebra ofLn · (ii) By Proposition 3.6.5, we can write

Let s be the least common multiple of nl - 1, ... , n r - 1, and n = s + 1. By the remark following Proposition 3.5.3 each of L n1 , .•• ,Lnr is a subalgebra of Ln. Thus, A E C(L n ). (iii) Let J be an ideal of A. Then a fortiori J is an ideal of the underlying lattice L(A). By our finiteness assumption, there is a E A such that J = (a 1 = {x E A I x ~ a}. By Corollary 1.5.6, a E B(A). It is easy to see that J is a maximal ideal of A iff there is

8.6. THE CARDINALITY OF F REEff.

173

an atom b of B(A) sueh that J = (-.b]. Let h : A - SI X ... X Sr an arbitrary isomorphism, where SI," ., Sr are all subalgebras of Ln. Let PI,'" ,Pr be the compositions of the canonical projections of L~ onto Ln with h. Trivially, Pi is a homomorphism of A into Ln. Furt her , for each i = 1, ... , r, it is easy to see that pi(A) = Si' Thus, for each i there is an atom bi in B(A) such that Ker(pd = (-.bi ]. It follows that (-.b l ] n ... n (-.br ] = (0], i.e., b1 V ... V br = 1. In conclusion, b1 , ... ,br exhaust all possible atoms of B (A), and PI, ... ,Pr are all possible homomorphisms of A into Ln. 0 An MVn-algebra A, with a distinguished subset Y of elements, is said to be a free MVn-algebra over the (generating set) Y , and is denoted by Freey, iff for every MVn-algebra Band every function f: Y - B, f ean be uniquely extended to a homomorphism of A into B. As in the ease of free MV-algebras (cf. Section 3.1), there is no danger of ambiguity in calling Free~ the free MVn-algebra over fi, many generators. Since for MV-terms p, q, p Ln = qL n iff pA = qA for each MVnalgebra A, arguing as in the proof of Proposition 3.1.4 we obtain: Proposition 8.5.4 For each natural number n 1, Free~ = Term(L n , fi,). 0

8.6

~

2 and cardinal

fi,

~

The cardinality of Free~

When fi, is a finite cardinal, say fi, = r ~ 1, L~: is finite, and has in fact nn elements. Therefore by the above Proposition 8.5.4 together with Corollary 8.5.3, Free~ is isomorphie to a direct product of a finite number of subalgebras of Ln. In this seetion, for each rand n we shall determine how many times each subalgebra of Ln appears in this direct product. Let Y = {Yb"" Yr} be a generating set of the free MV n-algebra Free~. There are n r functions from Y into Ln, denoted by el,' .. , enr. Let Si be the smallest subalgebra of Ln eontaining all ei (Yl), ... , ei (Yr), i = 1,2, ... , n r • Letting hi = hei be the unique homo mo rphi sm of T

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

174

into Ln extending ei, by Proposition 8.5.4, h l , .. . ,hnr is the list of all homomorphisms of Free~ into Ln; further, hi(Free~) = Si, for each i = 1, ... , n r . Thus, by Corollary 8.5.3(iii) we have Free~

= SI X ..• X Snr. = 1, ... , r, we define

Free~

For each i

= 1, ... ,nr and j

ei(Yj) Recall that Div(n),

aij

= (n - 1)'

(n 2: 2) denotes the set of all divisors d 2: 1 of

n. Given integers nl," ., nk

2: 1, we shall denote by gcd(nl" .. , nk)

their greatest common divisor. By Proposition 3.5.3 together with the remark following it, the set of subalgebras of Ln coincides with the set of algebras Ld+l, where d E Div(n - 1). Obviously, a subalgebra Si coincides with L 2 if and only if, for all j = 1, ... , reither aij = 0 or aij = n - 1. Further , for each d < n - 1 we have Si = Ld+l iff gcd(ail,"" air, n - 1) = (n - 1)jd. Since el,"" enr lists all functions of {Yl'" ., Yr} into Ln, for any given d E Div(n -1) there is an i such that aij = (n - 1)jd for all j = 1, ... , r. Such i also has the property that Si = L d+l . Therefore, all subalgebras of Ln are included among the Si 's, and we have

F reern = TI deDiv(n-l) LQ(n,r,d) d+l , r where a(n, r, 1) = 2 , and for each d > 1, a(n, r, d) is the number of rtuples (ail,' .. , air) E {O, 1, ... ,n -1 Y such that gcd(ail, ... ,air, n -1) = (n -1)jd. Let us fix n 2: 3, r 2: 1 and 1 =I d E Div(n-1). In order to compute the number a(n, r, d), we shall first introduce some notations. For each a and b in Div(n - 1), by writing

a-
= {a E Div(c) la -< cl,

8.6. THE CARDINALITY OF FREE:'

Min(c, n - 1)

175

= {a E Div(n - 1) I c -< a},

and H(c)

= {i E {I, ... , n r } I cE Div(ail) n ... n Div(airn.

Let us note that MaxDiv(c) = 0 Hf c = 1; moreover, Min(c, n-I) = 0 iff c = n - 1. Further, i E H(c) iff there are integers 0 :5 b1 , ... ,br such that bjc = aij :5 n - I,for all j = 1, ... , T. It follows that ~H(c) = (n~l + IY. We then have

{i E {I, ... , nr } I gcd(ail,""

air,

n-I n - 1) = -d-}

n-I

n-I

= H(-d-) \ (U{H(c) leE Min(-d-' n - In)· As a consequence, we can also write n-I n-I a(n, T, d) = ~H(-d-) - ~U{H(c) I cE Min(-d-' n -ln

= ~) _I)#X

nH(c),

cEX

where n-I o=J x ~ Min(-d-' n -

1).

For each nonempty X ~ Min(ndl, n - 1) let lcm(X) be the least common multiple ofthe elements of X. Then nCEX H(c) = H(lcm(X)), whence

~

n H(c)

cEX

=

(1

n-I (X) cm

+ Ir·

Note that a E Min(n~l, n - 1) iff n;l E MaxDiv(c). Defining now n-I

X* = { - leE X}, c

we get X ~ Min(ndl, n-I) iff X* ~ MaxDiv(d). We have just proved that 1c~(~) = gcd(X*), for all X=/: 0. Let us stipulate that gcd(0) = d.

176

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

Theorem 8.6.1 For any two integers n 2: 2 and r 2: 1 we have Free~

=

rr

dEDiv(n-l)

La:(n,r,d) d+l ,

where (_I)~X (gcd(X)

a(n, r, d) -

+ Ir.

0

X~MaxDiv(d)

From the above theorem, we get in particular Free~ = Lr, which is a weIl known characterization of the free boolean algebra with r free generators. Furt her , whenever n - 1 is prime we get

A simple counting argument shows that ijFreer = 4 = number of functions from L 2 into L 2 • As another example, while the number of functions from L 3 into L 3 is 27, a simple computation shows that ijFreef = 12. In the rest of this section we shall prove some interesting and less immediate consequences of Theorem 8.6.l. Let 'Jr be an MV-term in the variables Xl, ... ,Xn and suppose that S is a subalgebra of an MV-algebra A. From the definitions given in Section 1.4, it follows that for all al, . .. , ar E S

'JrA(al' ... ' ar ) = 'JrS(al' ... ' ar ) E S. Therefore, 'JrA(sr) ~ S. For short, we say that term functions preserve subalgebras. Since the subalgebras of Ln are precisely the algebras L d+ l , for d E Div( n-l), we can say that a function I: L~ -+ Ln preserves subalgebras if and only if 1 satisfies the following condition:

(*) For each r-tuple (n~l'···' n':l) E L~, il I(n~l'···' n':l) then b is a multiple 01 gcd(al' ... , ar , n - 1).

=

n~l'

Conversely, the next result shows that if a function I: L~ -+ Ln preserves subalgebras, then 1 is a term function. More precisely we have

8.7. BIBLIOGRAPHICAL REMARKS

177

Corollary 8.6.2 Given any arbitrary integers n 2: 2 and r 2: 1, let f: L~ ---jo Ln· Then fE Term(L n , r) iff f preserves subalgebras. Proof: Let 2 :5 d 1 < d2 < ... < ds = n - 1 display the divisors of n - 1 other than 1. Let (bil , ... , biT)' 1 :5 i :5 n T display all r-tuples in L~, where the indexes i are so chosen that • precedence is given to the 2T r-tuples that generate the subalgebra L 2 of Ln, • followed by the a(n, r, d2 ) many r-tuples that generate the subalgebra L d2 +1! • and so on. Let D be the subalgebra of L~ given by those functions satisfying condition (*). Let E be the product algebra rv T 11 l$i$s L o(n,T,d;) d; = erm (Ln, r ) .

It follows that the correspondence

is an isomorphism of D onto E. 0

8.7

Bibliographical remarks

The study of varieties of algebras is one of the main concerns of universal algebra. See, for instance, [104] or [150]. Proposition 8.1.1 originates from the early logico-algebraic Polish School (see [139]). Proposition 8.1.2 is due to Tarski [139, Theorem 20]. The notion of rank was introduced by Komori [130], who also proved the main results of Section 3. To prove Theorem 8.3.1 Komori introduced a special dass of ordered abelian groups. The proof given here, based on the properties of the variety of f-groups is borrowed from [50]. After Komori's dassification of all varieties of MV-algebras, considerable work has been done, by Gispert, Torrens, Panti and others,

178

CHAPTER 8. VARIETIES OF MV-ALGEBRAS

on their axiomatization, and on the classification and axiomatization of classes generated by MV-chains (see, e.g., [94], [95] [67], [68], [199]). The present equational characterization of all proper varieties of MValgebras is due to Di Nola and Lettieri. MVn-algebras were introduced by Grigolia [105], [106], who also gave their equational characterization. See also [212]. The variety generated by K 2 was considered in [7] and [65]. The description of the free MVn-algebra given in Theorem 8.6.1 was announced by Ant6nio Monteiro in the VI Congress of Mathematicians of Latin Expression, held in Bucharest during September 1969. The case n = 3 of equation (8.11) was established by A. Monteiro in his lectures delivered at the Universidad Nacional deI Sur, Bahia Blanca, in 1963. The case n = 4 was first obtained in [44]. Corollary 8.6.2. was first proved by McNaughton in [152]. Our proof is obtained by means of direct techniques from universal algebra. A brute force method to establish the nonisomorphism of finite algebras is to count their elements. For instance, as proved in [44), the 5-valued Moisil-Lukasiewicz algebra over one free generator has 192 elements, while, by (8.11), the free MV5-algebra over one generator has 300 elements.

Chapter 9 Advanced topics The first part of this chapter deals with disjunctive normal forms in the infinite-valued calculus of Lukasiewicz. We shall generalize the Farey-Schauder machinery of Chapter 3 to formulas in any number of variables. Disjunctive normal forms will be the key tool to prove McNaughton's theorem, generalizing the proof given in 3.2.8 for functions of one variable. We shall also discuss the relationships between normal form reductions and toric desingularizations, and the correspondence between MV-algebras and AF C· -algebras. Strengthening Corollary 4.5.3, we shall show that the tautology problem in the infinite-valued calculus is in fact co-NP-complete, thus having the same complexity as it boolean counterpart. We shall give a proof of Di Nola's representation theorem for all MV-algebras. With the possible exception of the first section, the pace in this chapter is usually fast er than in the previous chapters. Prerequisites ranging from l-group theory, polyhedral topology, NP-completeness theory, algebraic geometry, functional analysis, model theory, may be necessary for a complete understanding of the results in the following sections. In a final section appropriate references shall be given to the interested readers. 179 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

180

CHAPTER 9. ADVANCED TOPICS

9.1

McNaughton's theorem

For each (closed) n-dimensional simplex S S; [0, l]n with rational vertices we let (I)

VA, VI,. •. , V n

(11) 1 ~ di

(111)

Vij

be the list of vertices of S;

= least common denominator of the coordinates of Vi;

be the uniquely determined integers satisfying

with gCd(ViO, ... , Vi(n-I) , di ) = 1, where, as in the previous chapters, gcd denotes greatest common divisor; (IV) vfom E zn+! be the homogeneous coordinates of the vertices of S, (9.2) v?om

=

(ViO, . .. , Vi(n-I) ,

di ), i

= 0,1, ... , n;

x (n + 1) matrix whose ith row coincides with vfom. We say that the list of vertices va, VI, ... , V n has the positive orientation iff 1 ~ det(Ms).

(V) Ms be the (n

+ 1)

Following tradition, the set

(9.3) Ps

= {z E Rn+! I Z = Aov~om+ .. .+Anv~om,

°~ Ai < 1, Ai

E R}

shall be called the half-open parallelepiped of S. Definition 9.1.1 Adopt the above notation and terminology. Then S is said to be a unimodular simplex Hf det(Ms) = 1. By a unimodular triangulation of [0, l]n we mean a set W of closed n-dimensional unimodular simplexes with rational vertices, such that the union of all simplexes in W coincides with [O,I}n, and any two simplexes in Ware either disjoint or intersect in a common face. The following result strengthens Proposition 3.3.1:

9.1. MCNAUGHTON'S THEOREM

181

Theorem 9.1.2 Let 1 : [O,l]n -+ [0,1] be a McNaughton function, with its linear constituents PI! ... ,Pk. Then there is a unimodular triangulation U 01 [0, l]n such that lor each simplex S EU, 1 coincides with some Pj over S.

Proof: Let S be as given by Proposition 3.3.1. Adopting the above notation, let T be an n-dimensional simplex in S such that det(M T ) is maximum. Let VO, ••• , V n be the positively oriented list of vertices of T. If det(MT) = 1 we are done. Otherwise, det(M T ) > 1 and (MT )-1 is not an integral matrix. Let eo, ... ,en be the standard basis vectors of Rn+!: thus, eo = (1,0,0, ... ,0), el = (0,1,0, ... ,0), ... , e n = (0,0,0, ... ,1). We can safely assume that at least one of the coordinates of eo in the basis given by the vectors v~om, ... , v~om, is not an integer. Among all vectors that are obtainable by adding to eo linear combinations with integral coefficients of the vectors vfom, there is precisely one, denoted wERn+ 1 , (read: w = eo modulo PT), lying in the half-open parallelepiped PT' We have shown

°=J. w

=

(wo, .. ·, W(n-l), wn ) E zn+l n PT.

Let VT be the inhomogeneous correspondent of w, in symbols, VT = (wold, . .. , W(n-l)/d), where d = Wn = last co ordinate of w. Trivially, VT is a point of the n-simplex T, and is not one of its vertices. For any (n - l)-dimensional face F of T not containing VT, the point VT together with the vertices of F determines a new n-dimensional simplex [VT, F] ~ T. Let M = M[VT,F) be the associated matrix. Since the nonzero integral vector w belongs to the half-open parallelepiped PT, by (9.3), the determinant of M is an integer satisfying the inequalities (9.4) 1 ~ det(M) < det(M T ). Let c be the set of all n-dimensional simplexes of the form [VT' F], where F ranges over the (n - 1)-dimensional faces of T not containing the point VT. Replacing in S the simplex T by the new simplexes of c, we obtain a rational subdivision of S, Le., a set SI of n-dimensional simplexes with rational vertices such that (i) each simplex of S is a union of simplexes of SI! and

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182

(ii) any two simplexes in SI are either disjoint or they intersect in a common face. The set of vertices of (simplexes in) SI is the same as in S, with the only addition of VT. By (9.4), the number of simplexes R in SI such that det(M R ) = det(M T ) is decreased. In this way, proceeding by induction, we obtain a sequence of sets of n-dimensional simplexes

This sequence must terminate after a finite number Z of steps, yielding a set Sz = U of unimodular n-dimensional simplexes having the required properties. 0 Definition 9.1.3 Given a unimodular triangulation W of [0, l]n, and a vertex v of some simplex in W, let d be the least common denominator of the coordinates of v. Then the Schauder hat h v of W at v is the continuous function h v : [0, l]n ~ [0, 1] determined by the following conditions: (i) hv(v) = lid; (ii) hv(u) = for each vertex u E W different from v; (iii) hv is linear over each simplex U E W. We denote by 1iw the set of Schauder hats of W. The star of v, in the triangulation W-in symbols, star(v)-is the set of all n-dimensional simplexes of W having v among their vertices.

°

Proposition 9.1.4 Let W be a unimodular triangulation of [0, l]n, v a venex in W, and h v be its corresponding Schauder hat. Then we have (i) For any simplex S E star(v), let 9 be the linear polynomial coinciding with h v over S; then there are integers ao, . .. ,a(n-l), b such that for every x = (xo, ... ,X(n-l») E S,

(9.5)

hv(xo, ... , X(n-l»)

= aoxo + ... + a(n-l)X(n-l) + b = g(x);

(ii) h v E Free n . Proof: (i) Let Wo, ... ,Wn be a positively oriented list of vertices of S. By hypothesis, v = WI, for some l = 0, ... , n. Then the sequence of coefficients ao, . .. ,a(n-l), b coincides with the lth column of the inverse

9.1. MCNAUGHTON'S THEOREM

183

matrix (M S)-I; the latter is an integral matrix, by the unimodularity ofW. (ii) Let 91, ..• ,9q be the distinct linear polynomials given by (i). Our aim is to show that h v coincides, over U star (v), with a (V A)combination of the 9i 'so To this purpose, for any permutation U of the set {I, ... ,q}, we define the closed convex polyhedron Pu by (9.6)

Po'

= {x E [0, ljn I 90'(1) (x)

~ 9u(2)(X) ~ ••• ~ 9u(q) (x)}.

Let {} be the set of permutations 7r such that PTr is n-dimensional. Then

(9.7) v E

nP

[0, Ir =

Tr ,

TrEa

U PTr •

TrEa

For any J.1,11 E {} the polyhedra PJ.I. and P" are either disjoint, or else they intersect in a common face. Fix now an arbitrary permutation 7r E {}. Then, however we choose apermutation p E {} and i =1= j, the difference 9Tr(i) - 9T(0) will never vanish in the interior of Pp- Moreover, there is a unique index i Tr E {I, ... , q} such that (9.8) hv

= 9Tr(i.".) over

Again keeping

(9.9)

9 7r

7r

PTr

n

Ustar(v).

fixed, let us define the function 9Tr : Rn - R by

= 97r(1) A ... A 9Tr(i.".).

Claim. 9 Tr ~ hv over each simplex W E star(v). It is sufficient to prove the claim for all y in a suitably small open ndimensional ball B centered at v. By continuity, we can safely assume y to lie in the interior of Pp for so me permutation p E [1. By (9.8) there is an index i E {I, ... , q} such that hv = 97r(i) over Pp n B. If i ~ i Tr , the claim trivially follows by definition of 9 7r . If i > i 7r , there is a point x =1= y in the interior of PTr n B such that 9Tr(i) (x) < 9Tr(i.".)(X) = hv(x). We may assume x to be so close to v that the line segment [x, y] joining x and y lies in the interior of B n(U star(v)). Let X, Y E Rn+l be defined by X = (x, hv(x)) and Y = (y, hv(Y)). Let [X, Y] denote the line segment joining X and Y. Let 'fJ be the restriction of hv to [x, y]. Then'fJ is a continuous function, consisting of finitely many linear

CHAPTER 9. ADVANCED TOPICS

184

pieces. By our ass um pt ions about i and y, there is y' =f. y such that TJ coincides with g7r(i) over the interval N = [y', y] ~ [x, y]. Thus by our assumption about x, over the half-open interval N\ {y} the graph of TJ lies strictly below the segment [X, Y]. Among all W E [x, y] different from y and such that (w, TJ(w)) E [X, Y], there is a point z nearest to y (It is quite possible that z = x). Let j E {I, ... , q} be such that TJ( z) = g7r(j) (z), and TJ coincides with g7r(j) on a small open interval N' = [z, z'] ~ [z, y], for some z' =1= z. Then the restriction of the graph of g7r(j) to the half-open interval N'\ { z} lies strictly below the segment [X, Y]. Moving from z to y, we get g7r(j)(Y) < hv(Y). Moving from z to x, we get g7r(j) (x) > hv(x) = g7r(i,..) (x), whence j < i 7r . Thus, g7r ~ g7r(j) over all of Rn, which settles our claim. To conclude the proof, let gfl = V7rEfl g7r. By our claim, together with (9.8), since for each ()" E n, gfl = gU = hv over pun U star(v), then gfl = hv over star(v). Trivially, gfl V 0 = h v = Oover [0, l]n\ Ustar(v). Since gfl V 0 is a (V A)-combination of the gi U and the latter, by (i) and Lemma 3.1.9, are elements of Free n , the desired conclusion follows from Proposition 1.1.5. 0 Theorem 9.1.5 FOT each cardinal

K, the free MV-algebra FreeK, is given by the McNaughton functions over [0, Ir, with pointwise operations.

Praof: In the light of Propositions 3.1.4 and 3.1.8 it suffices to show that every McNaughton function f : [0, l]n ---+ [0,1] belongs to Free n . To this purpose, let U be as in Theorem 9.1.2. Let U E Qn be an arbitrary vertex of U, and let d be the least common denominator of the coordinates of u. Then f(u) = mu/d for some integer 0 ~ m u ~ d. Moreover, f is linear over each n-dimensional simplex of U. Let hu E 'Hu be as in Definition 9.1.3. Then the two continuous functions fand Eumuh u coincide over all of [0, l]n. Replacing sum by truncated sum EB, we have (9.10)

f = EB u

mu·hu

= h u EB h u EB ... EB h u (mu times).

Since, by Proposition 9.1.4, h u E Free n , then f E Freen. a required.

o

Remark: The set 1iu in the above proof is a DNF (Disjunctive Normal Form) reduction of f.

9.2. NONSINGULAR FANS AND NORMAL FORMS

9.2

185

Nonsingular fans and normal forms

This section requires familiarity with the so called vocabulary from toric varieties to fans. Writing in homogeneous coordinates the vertices of each simplicial complex arising in the disjunctive normal form of a McNaughton function, one obtains a sequence leading to a nonsingular refinement of the fan (a fan being a complex of simplicial cones, as defined below) corresponding to the linear domains of the proposition. This refinement process in turn amounts to aresolution of singularities for toric variety corresponding to the fan. In more detail, let So, SI, ... , Sz = U be a sequence of simplicial complexes over [0, l]n, as in the proof of Theorem 9.1.2. Each simplex T E Si has vertices Vo, ... , V n . The homogeneous counterpart of T is a simplicial cone, u T = (vo hom , ... , v n hom ), Le., the positive span in Rn+l ofvectors vohom, ... ,vnhom. As a vector in zn+l, each Vjhom is primitive, Le., minimal along its ray. The v/om are called the primitive generating vectors of u T . From each Si we obtain a simplicial fan .6.i , i.e., a complex of simplicial cones. The fact that Sz is a unimodular triangulation is equivalent to saying that the primitive generating vectors of each (n + l)-dimensional cone in .6. z form a unimodular matrix-for short, .6. z is a nonsingular fan. As is weIl known, every (nonsingular) fan.6. is canonically associated with a (nonsingular) toric variety XtJ., in such a way that the sequence .6.0 , ... ,.6.z corresponds to a desingularization X z of the toric variety X o. Thus, desingularizing a toric variety amounts to subdividing a simplicial complex into a unimodular triangulation, precisely as is done to compute DNF reductions of McNaughton functions. For the sake of definiteness, recalling Definition 9.1.3, let us give the following

Definition 9.2.1 A Schauder set in [0, l]n is a set of the form H = H u for some (necessarily unique) unimodular triangulation of [O,l]n. We say that H' is a one-step star refinement of H iff it is obtained from H as follows: (a) Pick a set S = {h 1 , ••• , h q } ~ Hand let h s = h 1 1\ ... 1\ h q ; (b) For each j = 1, ... , q replace hj by hj e h s ; (c) If =I hs put hs in H'.

°

186

CHAPTER 9. ADVANCED TOPIes

Trivially, H = H' if hs = 0 or if S is a singleton. When S has two elements and hs =J. 0 we say that the refinement is binary. Writing H = H u , and assuming 0 =J. h s , it follows that H' = HU', where U' is the unimodular triangulation obtained by starring U at the mediant point determined by the face S ofU. We say that H* is a star refinement of H iff it is obtained from H via a path H = Ho, H 1 , ... ,Ht = H*, where each Hi is a one-step star refinement of H i - 1 •

Theorem 9.2.2 Any two Schauder sets I and.c in [0, 1]n have a common star refinement. Proof: Readers familiar with toric varieties will recognize this statement as a reformulation of the strong form of Oda's conjecture. While for several years only the one-dimensional case of the conjecture was known to be true, (by Danilov's decomposition theorem) it appears that Morelli has finally settled the conjecture in the affirmative for the general case. 0 Thus, for any unimodular triangulation U we can explicitly construct H u starting from any Schauder set H w , and then applying only one (deduction) rule, namely the one-step star refinement. This method is more efficient than the inductive procedure given by Lemma 3.1.9.

Theorem 9.2.3 For any two Schauder sets Hand I there is a star refinement H* ofH such that every element ofI is a (truncated) sum of elements ofH*, as in the above formula (9.10). Moreover, all one-step star refinements leading /rom H to H* may be assumed to be binary. Proof: This is a consequence of the De Concini-Procesi theorem on elimination of points of indeterminacy in toric varieties. 0 In the particular case when n = 1, using the Hirzebruch-Jung continued fraction algorithm, one can compute, for every fan ß in the cartesian plane, the coarsest nonsingular subdivision of ß. In MValgebraic terms we have:

Corollary 9.2.4 Every set Q = {/I, ... , fk} ~ Freel has aleast DNF reduction H, i.e., a Schauder set H satisfying the following conditions:

• (i) Each fi is a (truncated) sum ofthe hats in H as in (9.10);

9.3. COMPLEXITY OF THE TAUTOLOGY PROBLEM

187

• (ii) Whenever a Schauder set.c is a DNF reduction o/Q, then.c is also a DNF reduction

0/ H.

o

9.3

Complexity of the tautology problem

In this section we require some familiarity with the theory of NPcompleteness. We shall consider the following problem: INSTANCE: A formula 4>. QUESTION: Is 4> a tautology in the infinite-valued calculus of Lukasiewicz ? We shall prove that the problem is co-NP-complete, Leo, the complementary problem of deciding whether a formula is not a tautology, is NP-complete. Cook's theorem states the co-NP-completeness of the tautology problem for the classical propositional calculuso We shall denote by 14>1 the number of occurrences of symbols in 4>. We shall also use the notation lxi for the absolute value of areal number x. This will never cause any confusion. Recalling Propositions 301.8 and 4.5.5, for each formula 4> = 4>(X l , . .• , X n ) its associated McNaughton function shall be denoted by f", = f",(Xl"" ,xn). For all x,y E [O,l]n, the one-sided derivative of f", at x along direction d = y - x is defined by /.' ( . d) - l' f",(x + Ed) - f",(x) '" x,

-

1m dO

E

0

Proposition 9.3.1 With the above notation, the directional derivative f~(x; d) is well defined, and we have the inequality (9.11) If~(x; d)1 where

Ildll

::; Ildll·I4>I,

denotes euclidean norm in Rn.

Proof: The existence of f~(x; d) follows by definition of McNaughton function. Inequality (9.11) is proved by induction on the number m of

CHAPTER 9. ADVANCED TOPICS

188

connectives occurring in . The basis m = 0 is trivial. For the induction step, if = -,'ljJ for some formula 'ljJ, then the desired conclusion immediately follows from the identity f I/> = 1 - f",· Finally, if = 'ljJ EEl X then, assuming without loss of generality that both 'ljJ and X have the same variables, the desired conclusion immediately follows by definition of truncated addition, upon noting that fl/> = f", EEl fx' 0 Corollary 9.3.2 Let P(XI, ... , Xn) = C + mIX + ... + mnXn be a linear polynomial with integer coejJicients c, ml,' .. ,mn' Let fl/>(XI,' .. ,xn) be the McNaughton function associated to a formula . Suppose fl/> eoineides with p over an n-simplex T ~ [0, l]n. Then we have

(9.12) max(lmll,···, ImnD ::; 11.

o Proposition 9.3.3 Let fl/>(XI, ... , Xn) be the MeNaughton funetion associated to a formula . Assume fl/> does not eoineide with the zero funetion over [0, l]n. Then there exists a point

x = (aI/b, ... , an/b)

E

[O,I]n

with ai, b E Z and 0 ::; ai ::; b (i = 1, ... , n) such that fl/>(x) > 0 and

o < b< 2(411/>1

2 ).

Proof: By Proposition 3.3.1 there is a finite number of distinct polynomials PI, .. . ,Pm with integer coefficients, and a finite number of ndimensional simplexes VI!"" V s , whose union is [0, l]n, and such that over each V j the function fl/> coincides with some Pi(j) ( i(j) = 1, ... , m). Suppose without loss of generality, x E VI and fl/>(X) > O. Then we can safely assurne x to be a vertex of VI' Therefore, the coordinates of x are all rational, say x = (aI/b, ... , an/b), for suitable integers ai and b with 0 ::; ai ::; b. Moreover, by the above corollary, x is the solution of a system of n linear equations in n unknowns, and each row has its coefficients ::; 211. Since, trivially, n ::; 11 then by Hadamard's inequality we conclude that the determinant /:). of this system satisfies the inequality Since b::; 1/:).1, the desired conclusion immediately follows. 0

9.3. COMPLEXITY OF THE TAUTOLOGY PROBLEM

189

Theorem 9.3.4 The tautology problem for the infinite-valued calculus is in the class co-NP.

Proof" A nondeterministie proeedure quiekly deciding if a formula

is not a tautology is as follows: Firstly, applying Proposition 9.3.3 to the function 1 - fljJ, guess a rational point

x

= (aI/b, ... , an/b)

°

E

[O,l]n

such that fljJ(x) < 1 and < b < 2(411/>1 2 ). Seeondly, for the purpose of eheeking that fl/>(x) < 1, write eaeh eoordinate ai/b as a pair of binary integers; let [ai] and [bI denote the number of bits of ai and b. Note that [ai] ~ [b] ~ 414>1 2 for all i = 1, ... , n; also note that, onee x is written down as a sequenee of pairs of binary numbers, its length [x] will satisfy the inequalities

Sinee the operations of negation and truneated addition do not inerease denominators, for some polynomial q : N ~ N the value fljJ(x) ean be eomputed by a deterministic Thring maehine within a number of steps ~ q(I4>I). 0 In order to prove that the tautology problem in the infinite-valued ealculus is eo-NP-hard, we prepare: Definition 9.3.5 For eaeh integer n 2: 1 and t 2: 2 we define the [0, 1]valued function fn,t by stipulating that for all x = (Xl, ... , X n ) E [O,l]n

fnAX) = {(Xl V -,xd 0 ... 0 (Xl V -'XI)} 0 ... ,

v

t times

'

t times

Further, for eaeh integer i 2: 1 the formulas 4>i, 'l/Ji,t, and Pn,t are defined by

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CHAPTER 9. ADVANCED TOPICS

(i) 4>i = Xi V..,Xi ; (ii) 'ljJi.t = 4>i 0 ... 04>i (t times); (iii) Pn.t = 'ljJl.t 0 ... 0 'ljJn.t. As an immediate consequence of the definition we have fn.t = A tedious but straightforward inspection yields the following

!Pn,t'

Lemma 9.3.6 Fix an enumeration of the vertices of the cu be [0, l]n. Let Vj be the jth vertex (j = 1, ... , 2n). Let

enumerate the edges oJ[O, ~]n adjacent to Vj. Por each i = 1, ... , n and t 2:: 2 let Yji be the point lying on edge Cji at a distance 1ft from Vj. Let Tj be the n-simplex with vertices Vj, Yjb"" Yjn' Then we have (i) fn,t{Vj) = 1; (ii) fn,t{Yji) = 0; (iii) fn.t is linear over each simplex Tj ; (iv) fn.t vanishes in [0, l]n outside U;:l Tj. 0 Lemma 9.3.7 Adopt the above notation. Let 4> = 4>(X1 , ••• ,Xn) be a formula. Let t = 14>1, and say without loss of generality, t 2: 2. Then 4> is a tautology in the boolean calculus iff Pn.t ~ 4> (i. e., "'Pn.t EB 4» is a tautology in the infinite-valued calculus.

Proof: For the nontrivial direction, by definition of implication, we must prove fn.t $ fl/l' With reference to Lemma 9.3.6(iv), the inequality holds over the set [0, l]n\ Uj Tj. So let us assume that for some j = 1, ... , 2n, fn.t(x) > fl/l(x) with x E Tj (absurdum hypothesis). By continuity we can safely assume x to be in the interior of Tj, whence in particular, x i= Vj' Let

9.4. MV-ALGEBRAS AND AF C*-ALGEBRAS

191

By our analysis, together with Proposition 9.3.1, for each point y lying in the interval [Vj, x] we have f~,t(Y; u)

:5

-t =

Further, f~(Y; u) ~

-14>1·

-14>1·

Thus, for all Y E [Vj, x], f~,t(Y; u)

Since by assumption, Lemma 9.3.6,

:5

-14>1 :5 f~(Y; u).

4> is a boolean tautology, fn,t(Vj)

ftf> = 1 on {O,1}n. By

= ftf>(vj) = 1.

Now, fn,t is linear on the interval [Vj, x]; on the other hand, ftf> is (continuous and) piecewise-linear on [Vj, x]. Thus ftf> ~ fn,t on [Vj, x], a contradiction. 0

Theorem 9.3.8 The tautology problem in Lukasiewicz infinite-valued calculus is co-NP-complete. Proof: We have just given a polynomial-time reduction of the boolean tautology problem into the tautology problem for the infinite-valued calculus. The desired conclusion is now an immediate consequence of Theorem 9.3.4 and the above Lemma 9.3.7, in the light of Cook's NPcompleteness theorem for the boolean satisfiability problem. 0

9.4

MV-algebras and AF C*-algebras

This section requires some familiarity with AF C" -algebras. Every C"-algebra A considered in this paper shall have a unit element 1A. By a projection p in A we mean a self-adjoint idempotent p = p. = p2. Up to isomorphism, the most general possible finite-dimensional C"algebra :F is a finite direct sum Md(l) + M d(2) + ... +Md(t) where Md(i) denotes the C"-algebra of all d(i) x d(i) complex matrices, for suitable 1 :5 d( i).

CHAPTER 9. ADVANCED TOPfCS

192

An approximately finite-dimensional (for short, AF C*-algebra) is the norm closure of the union of a sequence F 1 ~ F 2 ~ ..• of finitedimensional C*-algebras, all with the same unit, where each F i is a *-subalgebra of F i +1' For every AF C*-algebra A, two projections p, q E A are said to be equivalent Hf there exists an element v E A such that vv* = p and v*v = q. We denote by [P] the equivalence class of p, and by L(A) the set of equivalence classes of projections of A. The Murray-von Neumann order over L(A) is defined by

[P] ::;

[q] iff p is equivalent to a projection r such that rq

= r.

Elliott's partial addition, denoted +, is the partial operation on L(A) given by adding two projections whenever they are orthogonal. Then + is associative, commutative, monotone, and satisfies the following residuation property:

(*) For every projection pE A, among all classes [q] such that [P] + [q] = [lA] there is a smallest one, denoted ...,[p], namely the class [lA - p]. Theorem 9.4.1 For every AF C*-algebra A we have: (i) There is at most one extension 0/ Elliott 's partial addition to an associative, commutative, monotone operation $ over the whole L(A) having the above residuation property (*). Such extension $ exists iff L(A) is a lattice; (ii) Let K(A) = (L(A), [0],..." $). Then the map A f-t K(A) is a one-one correspondence between isomorphism classes 0/ AF C*-algebras whose L(A) is lattice-ordered, and isomorphism classes 0/ countable MV-algebras; (iii) In particular, up to isomorphism, the map A f-t K(A) is a oneone correspondence between commutative AF C*-algebras and countable Boolean algebras. The inverse correspondence is given by the map X t-+ C(X), the latter denoting the C*-algebra 0/ complex-valued continuous junctions over an arbitrary separable totally disconnected compact Hausdorff space X. 0

9.5. DI NOLA '8 REPRE8ENTATION THEOREM

193

Remarks. 1. Classes of AF C"-algebras A whose L(A) is a lattice include commutative, finite-dimensional, continuous trace, liminary with Hausdorff spectrum, as well as all AF C"-algebras with comparability of projections in the sense of Murray-von Neumann. 2. Intuitively, part (iii) in the above theorem suggests that MV-algebras are a noncommutative generalization of boolean algebras. Since, by Theorem 4.6.9, every countable MV-algebra is the Lindenbaum algebra of a theory e in the infinite-valued calculus of Lukasiewicz with denumerably many variables, any set ofaxioms for e is a presentation of a unique AF C"-algebra A e . The complexity of the word problem (in the sense of Theorem 4.6.10) of eis a faithful measure of the combinatorial complexity of A e . While most AF C"-algebras existing in the literature have polynomial time complexity, if the word problem e happens to be Gödel incomplete, then necessarily Ae has a nontrivial ideal. This shows the incompatibility of two equally imprecise and interesting conjectures: (a) that the C*-algebraic mathematizations of physical systems existing in nature should have no quotient structures, and (b) that Gödel incomplete AF C"-algebras might exist in nature. 3. Readers familiar with Grothendieck's group will recognize in the f-group Q(K(A)) the group Ko(A) equipped with the order induced by the image Ko(A)+ of the generating monoid of Ko(A).

9.5

Di Nola's representation theorem

This section requires familiarity with model theory.

Theorem 9.5.1 Up to isomorphism, every MV-algebra Bis an algebra 0/ [0, l]*-valued junctions over some set, where [0,1]* is an ultrapower 0/ [0,1], only depending on the cardinality 0/ B.

Proof" Let P( B) be the set of prime ideals of B. In the light of the Subdirect Representation Theorem 1.3.3, let us embed B into the MV-algebra II{B/I I I E P(B)}. For each prime ideal I of B, in the light of Theorem 7.1.7, let G(I) be the totally ordered abelian group with strong unit u(I) uniquely determined by the stipulation

194

CHAPTER 9. ADVANCED TOPICS

r(G(I),u(I)) C::! Bll. Let us embed G(I) into a totally ordered divisible abelian group K(I) with the same strong unit u(I). Let D(I) = r(K(I),u(I)). Then from Lemma 7.2.1 it follows that Bll is embeddable into D(I). Since any totally ordered divisible abelian group is elementarily equivalent to the additive group R of real numbers with natural order, it follows that D(l) is elementarily equivalent to the MV-algebra [0,1]. By Frayne's theorem in model theory, each D(I) is elementarily embeddable in a suitable ultrapower [0,1]*/ of [0,1]. The joint embedding property of first-order logic now yields an ultrapower [0,1]* (only depending on the cardinality of B), such that each MValgebra [0,1]*/ is elementarily embeddable into [0,1]*. Thus every Bll is embeddable into [0, 1]*, whence the desired conclusion immediately follows. 0

9.6

Bibliographical remarks

McNaughton's representation theorem 9.1.5 was first proved in [152], a few years before the advent of MV-algebras. The present proof, using unimodular triangulations and Schauder hats, was first given in [181]. Schauder hats over n-dimensional unimodular triangulations were introduced in [170]. Generalized Schauder hats are a standard tool in the theory of Banach spaces. Ewald's book [82] gives a full account of the vocabulary between fans and toric varieties. Also see [195], where toric desingularizations are discussed, along with the various results of Danilov, De ConciniProcesi and Hirzebruch-Jung mentioned in this chapter. See Morelli [162] for a solution of the strong form of Oda's conjecture. The relationship between toric desingularizations and the infinitevalued calculus of Lukasiewicz was first investigated in [183]. Direct applications to three-dimensional toric varieties are given in [5]. Theorem 9.3.8 was first proved in [167]. Any textbook on computational complexity theory contains a proof of Cook's NP-completeness theorem. A standard reference is [88]. See [77] and [101] for Elliott's theory, and for the role of Grothendieck's functor K o in connecting AF C"-algebras and partially ordered abelian groups. AF C"-algebras are a standard tool to give a math-

9.6. BIBLIOGRAPHICAL REMARKS

195

ematical description of spin systems in quantum statistical mechanics (see the references in [178]). The mutual relations between MV-algebras and AF C*-algebras were first investigated in [163]. Theorem 9.4.1 was proved in [190] building on [163]. Gödel incompleteness phenomena for AF C*-algebras are discussed in [163]. The fact that many weH known examples of AF C*-algebras are coded by polynomial-time theories in the infinite-valued calculus is noted in [166] and [177]. Theorem 9.5.1 is due to Di Nola, [63], [64]. The model theoretic machinery needed for its proof can be found in [42] and [158].

Chapter 10 Further Readings 10.1

More than two truth values

Lukasiewicz introduced many-valued logics in 1920. The history of studies of Lukasiewicz's original philosophical ideas and motivations is fairly long, and is definitely outside the scope of this book. Interested readers are referred to Wolenski's monograph [246], where Lukasiewicz's motivations are analyzed and his work on many-valuedness is presented in a wide perspective. In her essay [73] the author discusses manyvaluedness in the framework on nonclassicallogics. In their essay [204], Priest and Routley study Lukasiewicz logic from the viewpoint of paraconsistency. Paztig [200} discusses the relations between Lukasiewicz's ideas on many-valuedness and ideas in Chapter 9 of Aristotle's De Interpretatione. The short paper by Rosser [218] surveys the early stages of many-valued logic, and offers succinct historical and bibliographical remarks to an intended audience of physicists. The books [149], [30] and [227] contain English translations of papers by Lukasiewicz and Wajsberg. The problem of interpreting nonboolean truth values was considered by many people, including Lukasiewicz. Prior to the interpretation via Ulam game, agame theoretic interpretation of infinite-valued logic was given by Giles in [93], in the context of subjective probability. While no probability is involved in Ulam game, it is quite possible that an analysis of the Questioner's ''willingness to bet" in agame of Twenty 197 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000

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Questions with lies, links the two interpretations. See [92) for a first exploration. Other semantics for infinite-valued logic are given, e.g., in [223}, [236) and [238). Although many-valued logic was originally introduced in algebraic form by Lukasiewicz hirnself (see [139] and references therein), its definitive algebraization may be ascribed to Chang [36}, [38}, [40) who introduced MV-algebras and used the model theory of totally ordered abelian groups to prove the completeness of the Lukasiewicz axioms. The main ideas of Chang's completeness theorem are presented by Rosser in his early survey [219], and compared with the techniques used by Rose and Rosser for their own proof [216], "involving a frightening amount of detail" ([219, p.140]). In 1940 and 1941, Moisil introduced another dass of algebraic structures for his study of Lukasiewicz n-valued propositional calculi. These algebras are distributive lattices with a negation operation and some additional unary operations expressing modality (see [157)). While Moisil named these structures Lukasiewicz n-valued algebras, Rose proved that for n 2:: 5, it is impossible to define Lukasiewicz's n-valued implication building on Moisil's modal operations (see [44, p.2J). In the papers [46) and [47) it is shown how Moisil's constructions can be modified so as to provide adequate algebraic counterparts of Lukasiewicz n-valued propositional calculi. Independently of Lukasiewicz, Post in 1921 introduced his n-valued propositional calculi, see [203]. The algebraic counterparts of Post's calculi were described by Rosenbloom in 1942, [217) and named by hirn Post algebras of order n. In 1960 Epstein [79] investigated Post algebras from the lattice-theoretic viewpoint, (see [76) for Epstein's theory and its subsequent developments). In [44] it was shown that Post algebras of order n are obtainable from (Moisil's) Lukasiewicz algebras of order n by adding n - 2 constant operators. The monograph [28] presents a detailed study of Moisil algebras, Post algebras and (Moisil's) Lukasiewicz n-valued algebras. Among the texts concerned with many-valued Lukasiewicz logics let us mention Rosser-Thrquette [220}, Ackermann [1}, Rescher [206}, Gottwald [102], Bolc-Borowik [29}, and Malinowski [143]. The books by W6jcicki [244] and Hajek [112] contain chapters devoted to Lukasiewicz

10.2. CURRENT RESEARCH TOPICS

199

logic. Hähnle's monograph [108] is mainly devoted to automatic deduction in n-valued logics. Comprehensive bibliographies can be found in [157], [206] and [74] (also see the selected bibliography in [141].) The handbook chapters by Urquhart [237] and Panti [198] offer a survey of several many-valued systems. See [52] for a compact technical survey of MV-algebras and their neighbours.

10.2

Current Research Topics

We conclude with a list of active areas of current research in infinitevalued propositionallogic and MV-algebras. The relevant literature is rapidly growing, and several papers are under review at the time of writing this section.

10.2.1

Product

Generalized conjunction connectives over the unit real interval (also known as T-norms) are interesting objects of study from various viewpoints. The reader may consult the monograph by Butnariu and Klement [34] and the relevant chapters in Hajek's book [112] for background. One of the merits of T-norm theory is to show that a substantial portion of the expressive power needed for applications of infinite-valued logic to control theory, probability theory, and game theory with variable coalitions would be provided by a logic incorporating a product connective jointly with Lukasiewicz disjunction and negation. Many people are actively pursuing this line of research, including Di Nola, Dvurecenskij, Esteva, Georgescu, Godo, Leustean, Panti, Riecan. We refer, e.g., the reader to the papers [81], [207] and to Montagna's analysis [159] of the relationships between MV-algebras "with product" and various categories of lattice-ordered rings. A different approach is taken in [187], using tensor products-the latter perhaps being the bare minimum needed for if-then-else approximations of continuous real-valued functions. According to this approach, the Lukasiewicz calculus is as basic as groups are in algebraand (tensor) multiplication naturally appears as the fulfillment of the following desideratum: Having a "conjunction connective" that dis-

200

CHAPTER 10. FURTHER READINGS

tributes over the Lukasiewicz disjunction x EB y whenever x 0 y = 0 (Le., whenever x EB y = x + y). This distributivity law is a basic prerequisite to analyze and develop the expressive power needed for • the approximation of a (continuous, real-valued, control) function by means of a disjunction of pure tensors, • the definition, in every a-complete MV-algebra, of such notions as "independent events", "conditional", and "product of two observables", for furt her applications in MV-algebraic probability theory (see below).

10.2.2

States, Observables, Probability, Partitions

Introduced in [182] and [184], states are the MV-algebraic generalization of jinitely additive probability measures on boolean algebras. Their AF C* -algebraic counterparts are known as "tracial states". On the other hand, countably infinitary operations are needed for the development of MV-algebraic measure theory. Accordingly, a-complete MValgebras and a-additive states are systematicaHy used in the book by Riecan and Neubrunn [211]. As shown by Riecan and his School, many important results of classical probability theory based on a-complete boolean algebras and a-fields of sets have interesting MV-algebraic generalizations. One more example can be found in [187]. ' While the theory of a-additive MV-algebraic states is fairly weH understood, random variables (alias, observables) still lack a definitive systematization in the context of MV-algebras. A number of technical problems, also involving product and infinite distributive laws are posed by the theory of continuous functions of several (joint) MV-algebraic observables. (See [208], [209] and [210} for interesting positive results). A useful tool for understanding such observables is given by the MValgebraic generalization of the not ion of boolean partition [184], [185] and [188]. An MV-partition in A is a multiset of linearly independent elements of A whose sum equals 1. This definition makes perfect sense, by referring to the underlying Z-module structure of the unique latticeordered abelian group C with unit 1 given by r( C, 1) = A. The joint refinability of any two MV-algebraic partitions on an MV-algebra A depends on the "ultrasimplicial property" of its associated f-group C,

10.2. CURRENT RESEARCH TOPICS

201

in the sense that every finite set in G+ is contained in the monoid generated by some basis B S; G+, Le., a set B of positive elements that are independent in the Z-module G. After some partial results of Elliott, Panti, Handelman and others (see [171], [186], [192], [191] and references therein), recently Marra [145] has proved that every abelian f-group is ultrasimplicial.

10.2.3

Deduction

By contrast with finite-valued logic-and notwithstanding its rich algebraic structure-infinite-valued Lukasiewicz logic lacks a natural prooj theory. Currently used proof techniques spuriously range from variants of integer programming [108] and Fourier-Motzkin elimination [240} to the calculation of level sets of McNaughton functions [189], [2]. The paper [4] is a first attempt to introduce an analytic calculus for the infinite-valued propositional logic of Lukasiewicz. Tight estimates are given for the complexity of the consequence relation, thus strengthening earlier results in [167]. Methodologies for automated deduction in infinite-valued Lukasiewicz logic are currently investigated by several people, including Aguzzoli, Baaz, Ciabattoni, Escalada Imaz, Fermüller, Hähnle, Lehmke, Manyia Serres, Olivetti, PauHk, Salzer, Vojtas, Wagner. See for instance [239], [137], [109], [80], [189], [2], [3], [240], the survey paper [111] and the handbook chapter [9]. Various not ions of literal, clause, resolution are being considered, and various types of deduction pro cedures for "easy cases" are implemented, by analogy with the Horn case and 2-CNF case in the classical propositional calculus. Altogether, much work is still to be done before proof theory and automated deduction in infinite-valued propositional logic reach a mature stage.

10.2.4

Further constructions

MV-algebraic coproducts (= MV-algebraic free products) are considered in [172} and by Di Nola, Lettieri in [66]. In [168] it is proved that the amalgamation property holds in the variety of MV-algebras. This result is generalized in [67].

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For automorphisms of free MV-algebras see [71]. A moment's reflection shows that the automorphism group of the free MV-algebra with one generator consists of precisely two elements: identity and rotation of McN aughton functions around the vertical li ne through the point (1/2, 0). By contrast, already the automorphism group of the free MV-algebra with two generators is highly nontrivial. Order-theoretic and topological properties 0/ the set 0/ ideals of MValgebras, as weH as various kinds of representations, including sheaf representations, are studied in [221, [57], [17], [16], [72}, [146], [84J, [147], [148], [205], [116], [117], [118], [120], [70J. The MV-algebraic reformulation of a long-standing open problem in f-group theory asks for a purely topological characterization of the possible spaces of prime ideals of MV-algebras with the natural (huH-kernel) topology. The order-theoretic characterization was given in [56]. The relations between MV-algebras and various structures, notably distributive lattices, ordered monoids and f-groups, are investigated in

[213], [214J, [31], [32], [13], [173], [63], [35], [96], [196], [115], [19], [180], [78J and [59]. Special classes of MV-algebras are considered in [7], [21], [20], [54], [99], [100], [114], [131], [133], [134], [180], [85], [75] [18].

A classification of the universal classes generated by certain totally ordered MV-algebras is given in [95]. For subvarieties, quasi-varieties and related topics, see [215], [230], [248], [27], [83], [87], [94J. In this book, finite-valued Lukasiewicz logics have been considered only via their Lindenbaum algebras, namely Grigolia's MV n-algebras, when dealing with the classification problem for subvarieties of MV-algebras. For more information on the algebras and proof theory of such systelJls see, for instance, [160], [161], [47], [48], [157J, [8], [10], [11], (122), [173],

[226J.

As already noted, infinite-valued "first-order" not ions have not been considered at all. A rapidly growing literature is concerned with the important problem of giving infinite-valued generalizations of the classical notions of point, set, cartesian product, union, subset, relation, function, equality, model, quantifier. For various constructs and attempts in this direction see, e.g., [41], [222], [90], [48}, [110], [112J and references therein.

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Index 01=, the tautologies, 81 A/F, the quotient of A by F, 87 a « b, a is infinitely smaller than b, 150 Ho G, lexicographic product, 152 (W), the ideal generated by W, 13 n.x, the n-fold truncated addition of x, 33 lai, the equivalence class of formula a, 93 r functor, 34 el-, the provable formulas from e, 88 el=, the semantic consequences of e, 81

Adaptive search, 109 Adjoint functor, 139 AF C"'-algebra, 192 commutative, 193 continuous trace, 193 finite-dimensional, 193 liminary, 193 with comparability of projections, 193 Aguzzoli, S., 194, 201 Algebra AF C*-, 192 BCK-, 29 C*-, 191 CN-, 30 De Morgan, 137 Kleene, 137 Lindenbaum, 94 1-, 30 MV-, 7 Moisil-Lukasiewicz, 198 Post, 198 So, 30 Wajsberg, 83 Alphabet, 21 of Lukasiewicz calculus, 79 Amalgamation property, 202 Ambrosio, R., 202 Analytic calculus, 201 Answer in Ulam game, 104 Archimedean element, 32 of an MV-algebra, 115 Aristotle's, De Interpretatione, 197 Atom in a poset, 122 in an MV-algebra, 132 Atomic MV-algebra, 132 Atomless MV-algebra, 132 Automated deduction, 201 Automorphisrns of free MV-algebras, 202 Axiom, 87

C(Je), the variety generated by Je, 158

Cont(X), 65 Div(n), the divisors of n, 169 Form, the set of formulas, 79 Free" , 53 H n , Komori chain of the second type, 160 I(A), the ideals of A, 14 Injinit(A), the infinitesimals of A, 73 Js, 65 Ker(h), the kernel of h, 13 K n , Komori chain of the first type, 160 .c(S), the Lindenbaum algebra of S, 99 Ln, the n element Lukasiewicz chain, 70 M(A), the maximal ideals of A, 14 "P(A), the prime ideals of A, 14 Q, the additive o-group of rationals, 33 R, the additive o-group of reals, 33 Rad(A), the radical of A, 72 u-additive state, 200 u-complete MV-algebra, 129 u-field of sets, 200 Theo, the lattice of theories, 97 U n , the variety of n-bounded MV-algebras, 160 Var(a), the variables in a, 79 VJ, 65 Z, the additive o-group of integers, 33 Abelian monoid, 8 Absolute equation, 107 value, 32 Ackermann, R., 198

Baaz, M., 201, 202 Basis, 57 BCK-algebra, 29

225

INDEX

226 Belluce, L. P., 156, 202 Berlekamp, E. R., 109 Birhkoff, G., 19, 137, 143 Blok, W. J., 202 Boolean element, 25 product, 125 space, 120 Bosbach, B., 30, 138, 202 Brick, 30 Buff, H. W., 30 Butnariu, D., 30, 199 C*-algebra, 191 Cancellation, 37 Cantor set, 120 Casari, E., 202 Cauchy, A., 56 Chang, C. C., 20, 29, 44, 49, 76, 82, 101, 156, 198, 202 l-group, 42 Ciabattoni, A., 201 Classical propositional calculus, 78 Clopen set, 119 CN-algebra, 30 Communication with feedback, 109 Complement, 25 Complemented element, 25 Complete lattice, 129 MV-algebra, 129 Completely distributive lattice, 134 MV-algebra, 134 Completion of an MV-algebra, 138 Cone, simplicial, 185 Congruence relation, 15 Conjunction of states in Ulam game, 106 Connective bi-implication, 79 implication, 78 negation, 78 product, 199 Consequence semantic, 80 syntactic, 88 Coproduct, 202 Covers, 174 Danilov's decomposition theorem, 186 De Morgan algebra, 137 Decidability of word problem, 95 Decision problem, 77 Deduction theorem, 98 Desingularization of a toric variety, 185

Di Nola, A., 156, 178, 193, 195, 199, 201, 202 Direct product, 19 Directly indecomposable MV-algebra, 123 Disjunctive normal form, 184 Distance function, 15 Distributive lattice, 24 Divisible o-group, 194 DNF reduction, 184 Dvurefenskij, A., 199 Elliott, G. A., 201 c1assification theory, 194 partial addition, 192 Embedding, 13 Enveloping group, 40 Epstein, G., 198 Equation, 22 Equivalence logical, 96 of projections, 192 Equivalent formulas, 80 good sequences, 40 Error-correcting code, 109 Escalada Imaz, G., 201 Esteva, F., 199 Fan, 185 nonsingular, 185 simplicial, 185 Farey partition, 56 sequence, 56 Farkas lemma, 49 Fermüller, C., 201, 202 Ferreirim, I. M. A., 202 Filipoiu, A., 202 Filter implicative, 86 in a lattice, 114 Font, J. M., 30, 101 Formula 2-CNF, 201 Horn, 201 equivalent, 80 provable, 88 satisfied by a valuation, 80 Fourier-Motzkin elimination, 201 Frayne's embedding theorem, 194 Free MV-algebra, 53 MVn -algebra, 173 product, 202 Gaitan, H., 202

INDEX Generated ideal, 13 subalgebra, 8 Generating set of a subalgebra, 8 Georgescu, G., 138, 199, 202 Giles, R., 197 Gispert, J., 178,202 Giuntini, R., 202 Gluschankof, D., 137, 202 Gödel incomplete AF C"'-algebra, 193 the6ry, 193 Godo, L., 199 Good sequences, 34 equivaIent, 40 natural order of-, 38 sum of-, 36 Gottwald, S., 198 Grothendieck's group Ko, 193, 194 Grigolia, R., 178, 202 Group abelian, 31 divisible, totally ordered, 194 enveloping, 40 lattice-ordered, 31 partially ordered, 31 torsion-free abelian, 45 totally ordered, abelian, 31 ultrasimplicial, 201 Hähnle, R., 199,201 Hajek, P., 198, 199, 202 Half-open parallelepiped, 180 Hay, L., 76, 101 Hirzebruch-Jung algorithm, 186 Höhle, U., 202 Homogeneous coordinates, 180 Homomorphism, 12 injective, 13 kernel of-, 13 natural, 16 of i-groups, 34 surjective, 13 unital, 34 Hoo, C. S., 30, 138, 202 Horn formula, 201 Hyperarchimedean MV-algebra, 116 Hyperplane, 62 Ideal

dual in a lattice, 114 generated by a subset, 13 in a lattice, 112 in an MV-algebra, 13 maximal, 14 maximal stonean, 124

227 minimal prime in a lattice, 113 nilpotent, 151 prime, 13 prime in a lattice, 112 principal in a lattice, 112 principal, 13 proper, 13 proper in a lattice, 112 stonean in a lattice. 115 Ideals poset of prime-, 202 spectral space of prime-, 202 Idempotence law, 26 Implication connective, 79 Implicative filter, 86 Incompatible pair of states, 107 state of knowledge, 106 Infinite-vaIued calculus of Lukasiewicz, 77 Infinitely small, 73 Infinitely smaller, 150 Infinitesimal, 73 Inhomogeneous coordinates, 57 correspondent, 181 Initial state of knowledge, 106 Injective homomorphism, 13 Iorgulescu, A., 202 Iseki, K., 29 Isomorphism, 13 Iturrioz, L., 202 Jakublk, J., 138 Joint embedding property, 194 Keisler, H. J., 202 Kernel of a homomorphism, 13 Kleene algebra, 137 Klement, E. P., 30, 199 Komori, Y., 30,101,167,177 Lacava, F., 30, 138,202 Lattice complete, 129 completely distributive, 134 distributive, 24 ideal of, 112 Lattice-ordered group, 31 Lehmke, S., 201 Lettieri, A., 156, 178, 201, 202 Leustean, I., 199, 202 Lexicographic product, 152 i-group, 31 homomorphism, 34 term, 43 of an MV-algebra, 42

228 Lindenbaum algebra, 94 of a theory, 99 Linear constituent, 62 Logical equivalence, 96 Lukasiewicz, J., 29,78, 102, 103, 178, 197 axioms, 49 calculus, 77 chain, 122 connectives, 78 finite-valued calculi, 78 Malinowski, G., 198 Mangani, P., 29 Manyili Serres, F., 201 Marra, V., 201 Martfnez, N., 202 Maximal ideal of an MV-algebra, 14 stonean ideal, 124 Maximum satisfiability problem, 109 McNaughton, R., 62,75, 178, 194 function, 54 functions, of one variable, 62 representation theorem, 184 Mediant, 56 Menu, J., 30 Meredith, C. A., 102 Minimal prime ideal of a lattice, 113 Model completion of an MV-algebra, 138 Modus ponens, 88 Moisil, G., 178, 198,202 Moisil-Lukasiewicz algebra, 198 Monoid, zero-law, 37 Montagna, F., 199 Monteiro, A., 178, 202 Morelli's proof of Oda's conjecture, 186 Multiplicity of a Schauder hat, 59 Murray-von Neumann order of projections, 192 MV-algebra, 7 u-complete, 129 archimedean element, 115 atom, 132 atomic, 132 atomless, 132 chain, 10 completely distributive, 134 complete, 129 completion, 138 coproduct, 202 directly indecomposable, 123 equation, 22 free, 53 free product, 202 homomorphism, 12 hyperarchimedean, 116

INDEX idempotent, 26 maximal stonean ideal in-, 124 natural order, 10 n-bounded, 160 nontrivial, 8 one-generated hyperarchimedean, 119 one-generated free, 62 partition, 200 perfect, 152 probability measure, 200 product, 19 quotient, 16 radical, 72 semisimple, 72 separating, 66 simple, 70 state, 200 subalgebra, 8 subdirect product, 19 subterm, 21 tensor product, 199 term, 21 totally ordered, 10 valuation, 79 variety, 157 without linear factors, 135 MVn -algebra, 172 Natural homomorphism, 16 order between states of know ledge, 107 order in an MV-algebra, 10 order of good sequences, 38 n-bounded MV-algebra, 160 Negation connective, 79 in Ulam game, 107 Negative answer, 106 part, 32 Neubrunn, T., 200 Nilpotent ideal, 151 Node of a function, 59 Non-isomorphic simple MV-algebras, 148 Nonsingular fan, 185 Oda conjecture, strong form, 186 o-group, 31 Olivetti, N., 201 One-generated hyperarchimedean MV-algebra, 119 One-step star refinement, 185 Ono, H., 202 Order natural, 31 translation invariant, 31

229

INDEX underlying, 31 Order unit, 32 Panti, G., 49, 178, 199, 199, 201, 202 Partially ordered abelian group, 31 Parsing sequence, 21 Partition in an MV-algebra, 200 Pasquetto, M., 49 Patzig, G., 197 Paulfk, L., 201 Pavelka, J., 30 Perfect MV-algebra, 152 Polyhedron, 63 Poset of prime ideals, 202 Positive answer, 106 cone, 31 orientation, 180 part, 32 Post, E., 198 algebra of order n, 198 Precedence laws, 9 Priest, G., 197 Priestley, H. A., 202 Prime ideal of a lattice, 112 ideal of an MV-algebra, 13 vector, 185 Principal ideal, 13 ideal of a lattice, 112 Probability measure on an MV-algebra, 200 Product in infinite-valued logic, 199 Projection, 52 function, 19 in a C'"-algebra, 191 Proof in the infinite-valued calculus, 88 Proper equational dass, 157 ideal, 13 ideal of a lattice, 112 variety, 157 Propositional formula, 79 variable, 79 Provable formulas, 88 = tautologies, 95 Thring enumeration of-, 89 Pseudocomplement, 133 Question in Ulam game, 104 Quotient algebra, 16 Radical of an MV-algebra, 72 Ramana Murty, P. V., 30 Rank of an MV-chain, 158

Rational subdivision, 181 vertex, 63 Rescher, N., 198 Residuation in Elliott's addition, 192 Resolution, 201 Ri~an, B., 199, 200 Rodr(guez, A. J., 30, 76, 101, 137 Romanowska, A., 30,202 Rose, A., 49, 75, 101, 198, 202 Rosenbloom, P. C., 198 Rosser, J. B., 49,75, 197, 198 Routley, R., 197 Saeli, D., 138 S-algebra, 30 Salzer, G., 201 Schauder hat, 182 multiplicity, 59 of a Farey partition, 58 Schauder set, 185 Schwartz, D., 202 Search space in Ulam game, 103 Semantic consequence, 80 equivalence, 81 Semisimple MV-algebra, 72 Separating MV-algebra, 66 Sessa, S., 138, 202 Sharper state of knowledge, 107 Simple MV-algebra, 70 Simplex, 63 unimodular, 180 Simplicial cone, 185 fan, 185 Spectral space of prime ideals, 202 Stachniak, Z., 202 Star in a triangulation, 182 refinement, 186 State of knowledge in Ulam game, 106 on an MV-algebra, 200 Stone, M. H., 137 Stonean ideal in a lattice, 115 String of symbols, 21 Strong order unit, 32 Subalgebra generated by a subset, 8 Subdirect product, 19 of i-groups, 143 Subsystem, 60 Subterm, 21 Sum of good sequences, 36 Surjective homomorphism, 13 Symbols of an alphabet, 21

230 Symmetrie difference, 15 Synta.ctic consequence, 88 equivalence, 92 Thnaka, S, 29 Tarski, A., 101, 135, 138, 177 Tautology, 80 Tensor product of MV-algebras, 199 Term function, 22 of an l-group, 43 Theorem r is fuH, faithful and dense, 146 r preserves quotients, 148 MV = C([O, 1», 158 Chang's completeness, 44 Chang's subdirect representation, 20 Cook, 187 Oanilov, 186 Oe Concini-Procesi, 186 Oi Nola, 193 Frayne, 194 Komori 's dassification of MV-varieties, 169 McNaughton, 184 McNaughton-, for Freel, 62 Morelli, 186 Stone representation, 82 completeness in Ulam game, 108 completeness of the infinite-valued calculus, 81 co-NP-completeness of tautology problem, 191 decidability of the word problem, 95 deduction, 98 finitely axiomatizable theories are decidable, 100 categorical equivalence, 146 on complete MV-algebras, 134 on finite-valued MV-algebras, 168 on Free~, 175 on hyperarchimedean MV-algebras, 117 on perfect MV-algebras, 155 on principal ideals, 69 on rank n nonsimple MV-chains, 167 on semisimple MV-algebras, 75 on simple MV-algebras, 70 on unimodular refinement, 180 principal MV-quotients are semisimple, 75 spectral representation, 66 Theory, 97 T-norm, 199 Tokarz, M., 202 Toric

INDEX desingularization, 185 variety, 185 vocabulary, 185 Torrens, A., 30, 101, 137, 178, 202 Totally ordered abelian group, 31 Tra.czyk, T., 30, 202 Translation invariance, 31 Triangulation, 63 star in a-, 182 unimodular, 180 Truth-value, 105 Turing decidability of absolute equations, 108 decidability of word problem, 95 enumeration of nontautological formulas, 81 enumeration of provable formulas, 89 Turquette, A., 198 Turunen, E., 138 Two-valued calculus, 78 Ulam, S., 103, 197 Ulam game of Twenty Questions, 103 absolute equation, 107 answer, 104 incompatible pair of states, 107 incompatible state, 106 initial state, 106 natural order, 107 negation, 107 negative answer, 106 positive answer, 106 question, 104 search spa.ce, 103 sharper state, 107 state of knowledge, 106 truth-value, 105 Ultrapower of [0, 1], 194 Ultrasimplicial group, 201 Underlying lattice of an MV-algebra, 10 Uni modular simplex, 180 triangulation, 180 Unique readability of formulas, 80 of l-group terms, 43 of MV-terms, 21 Unit interva1 in an l-group, 34 Unitall-homomophism, 34 Universal dass of an MV-chain, 202 Urquhart, A., 199 Valuation, 79 in [0,1], 81 Variety, 157 generated by, 158

INDEX Vertices of a polyhedron, 63 Vojtas, Po, 201 Wagner, Ho, 201 Wajsberg, Mo, 49 algebra, 83 Weak boolean product, 124 W6jcicki, R., 76, 101, 198 Wolenski, Jo, 197 Word, 21 Word problem decidability of, 95 for finitely presented algebras, 100 Wozniakowska, Bo, 202 Yutani, Ho, 29 Zero law in a monoid, 37 Zeroset, 65

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