Zbl-didaktik-vasantha1-2

Zentralblatt fur der Didkatik der Mathematik, Germany 2002f.04934 Vasantha Kandasamy, W.B.(Indian Inst.of Tech., Madras (India)) Smarandache Semigroups. (English) [Book] Rehoboth, NM: American Research Press. 2002. 93 p. [ISBN 1-931233-59-4] Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B$\subset$A which is embedded with a stronger structure S. Thus, as a particular case: A Smarandache Semigroup is a semigroup A which has a proper subset B$\subset$A that is a group (with respect to the same binary operation on A). The book assumes the reader to have a good background in group theory. It has seven chapters. The first chapter on preliminaries give some important notions and concepts, which are used in this book. Chapters 2 and 3 gives most of the basic concepts on group theory and results in group theory which have been used in this text to study Smarandache notions in groups or Smarandaches semigroups. The text does not in any way claim completeness in giving the properties of groups. Chapter 4 starts with the definition of the Smarandache semigroup and gives some interesting properties of Smarandaches semigroups. Several examples and problems are given. Chapter 5 makes use of the newly defined and special types of Smarandache semigroups in proving or disproving the classical theorems or analogs of the classical theorems. Chapter 6 is a mixture of both Smarandache notions on groups and the study of properties of Smarandache semigroups. The final chapter (7), a special attraction to researchers and algebraists, is a list of open research problems. MESC: *H40 Groups, rings, fields E60 Sets. Relations. Set theory Keywords: Group Theory; Algebra; Semigroups; Groups; Foundations of Mathematics; Mathematical Structures; Set Theory; Algebraic Structures; Abstract Spaces \par Gruppentheorie; Algebra; Halbgruppe; Gruppe; Grundlagen der Mathematik; Mathematische Struktur; Mengenlehre; Algebraische Struktur; Abstrakter Raum

Zentralblatt fur der Didkatik der Mathematik, Germany 2002f.04932 Vasantha Kandasamy, W.B.(Indian Inst.of Tech., Madras (India)) Groupoids and Smarandache Groupoids. (English) [Book] Rehoboth: American Research Press. 2002. 113 p. [ISBN 1-931233-61-6] Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B$\subset$A which is embedded with a stronger structure S. Thus, as a particular case: A Smarandache Groupoid is a groupoid G which has a proper subset S$\subset$G such that S under the operation of G is a semigroup. The book aims to give a systematic development of the basic non-associative algebraic structures viz. Smarandache groupids. The book assumes that the reader should have a good background of algebraic strcutureslike semigroup, group etc. and a good foundation in number theory. Chapter 1 recalls the basic notations and some important definitions used in this book. Chapter 2 introduces almost all concepts, most of them new, to groupoids in general. Chapter 3 introduces four new classes of groupoids using the set of modulo integers $Z\sb n$, n$\ge$3 and n$<$$\infty$. In this chapter, several number theoretic techniques are used. Chapter 4 starts with the definition of Smarandache groupoids. All properties introduced into groupoids are studied in the case of Smarandache groupids. Several problems and examples are given in each section to make the concept easy. In chapter 5 conditions for the new classes of groupids built using $Z\sb n$ to contain Smarandache groupids are obtained. Chapter 6 gives the application of Smarandache groupoids to semi automaton and automaton, that is to finite machines. The final chapter on research problems is the major attraction of the book giving several open problems about groupoids. MESC: *H40 Groups, rings, fields E60 Sets. Relations. Set theory Keywords: Group Theory; Groups; Semigroups; Automata Theory; Algebra; Abstract Spaces; Mathematical Structures; Foundations of Mathematics; Algebraic Structures; Set Theory \par Gruppentheorie; Gruppe; Halbgruppe; Automatentheorie; Algebra; Abstrakter Raum; Mathematische Struktur; Grundlagen der Mathematik; Algebraische Struktur; Mengenlehre