Chapter Iii

Chapter-III MODERN ALGEBRA

Contents: 3.1

Binary Operations

3.2

Properties of Binary operations

3.3

Semigroup

3.4

Monoid

3.5

Group

3.6

Groupoid

MODERN ALGEBRA 3.1 Binary Operations: A binary operation is simply a rule for combining two objects of a given type, to obtain another object of that type. Through elementary school and most of high school, the objects are numbers, and the rule for combining numbers is addition, subtraction, multiplication or division. Binary operation on a set S. A binary operation on a set S is a rule which assigns to each ordered pair a,b of elements in S a unique element c = ab. Closure. A set S is closed with respect to a binary operation if and only if every image ab is in S for every a,b in S. 3.2 Properties of Binary operations: Commutative operation: A binary operation on a set S is called commutative if xy = yx for all x,y in S. Associative operation: A binary operation on a set S is called associative if (xy)z = x (yz) for all x,y,z in S. Distributive: Let S be a set on which two operations ∙ and + are defined. The operation ∙ is said to left distributive with respect to + if a ∙(b + c ) = (a∙b) + (a∙c) for all a,b,c in S and is said to be right distributive with respect to + if (b + c)∙a = (b∙a) + (c∙a)

for all a,b,c in S

Existence of identity elements and inverse elements: Identity element: A set S is said to have an identity element with respect to a binary operation on S if there exists an element e in S with the property ex = xe = x for every x in S. Inverse element: If a set S contains an identity element e for the binary operation , then an element b S is an inverse of an element a S with respect to if ab = ba = e . Note. There must be an identity element in order for inverse elements to exist.

Theorems : Theorem 1. A set S contains at most one identity for the binary operation . An element e is called a left identity if ea = a for every a in S. It is called a right identity if ae = a for every a in S. If a set contains both a left and a right identity, they are the same. Theorem 2. An element of a set S can have at most one inverse if the operation is associative.

In general, in regular algebra, when one multiplies several real numbers together, a product of several numbers is assumed to have a particular value independent of how the multiplications are performed (i.e. where parentheses are placed): x1x2x3 ... xn = x1(x2x3)(x4 ... xn) = (x1x2)(x3x4)(x5 ...xn) or, in terms of numbers, 5∙3∙8∙7∙3∙9 = 5(3∙8)(7∙3)9 = (5∙3)(8∙7)(3∙9) = ... The product is unique, independent of the placing of the parentheses. This rule is true in the case of the multiplication of real numbers. It is not, however, in general true with an arbitrary operation . Under what conditions is it true? It is true on a closed set S which has an operation which is associative. The operation of multiplication on the real numbers is associative and so this product is unique for the multiplication of real numbers. Theorem 3. Let a set S be closed with respect to an associative binary operation . Then the products formed from the factors

, multiplied in that order, and with the

parentheses placed in any positions whatever, are equal to the general product

.

Note that the theorem refers to the grouping -- the order of the numbers remains the same. The concept of a binary operation is a very general one, and need not be restricted to sets of numbers. In fact, an operation can be specified on any finite set simply by presenting a table that shows how the operation is performed, when you are given two elements of the set. For example, consider the set table:

and an operation, denoted by *, defined by the following

We interpret this operation table in much the same way that we would interpret an addition table. Using the operation symbol * as we would use + to mean addition, the table shows us, among other things, that

and so on. The table summarizes 16 such calculations, telling us how to combine each of the four elements of A with each of the four elements. Not All Operations Have the Same Properties You should be familiar with various properties of the arithmetic operations on numbers. Addition of numbers, for instance, is a commutative operation -- meaning that for all numbers x and y. The operation on the set A defined by the operation table above, however, is not commutative, and there are several instances of this lack of commutativity. For instance, since the table shows that . In general, commutativity is a property of an operation, so it takes only one instance of lack of commutativity to spoil that property for the operation. It is easy to check whether an operation defined by a table is commutative. Simply draw the diagonal line from upper left to lower right, and then look to see if the table is symmetric about this line. In the illustration below, we see a lack of symmetry: the table entries colored yellow do not match, and the table entries colored blue do not match.

Either one of these mismatches would be sufficient to make the operation non-commutative. Addition of numbers is an associative operation, meaning that for all numbers x, y and z. To check to see whether the operation * defined above is associative, however, is a somewhat tedious task. We would need to compute all combinations of the form in two ways -- once as shown, and then again in the form -- and then check to see that they are equal. This must be done for each selection of elements to fill the placeholders. In the case of a 4-element set such as A above, there are choices of the elements to be used, and each must be computed in two ways. Thus, to verify that a binary operation on a 4-element set is associative, we would have to do 128 computations! There is no easy shortcut as there is for checking commutativity.

On the other hand, if a given operation fails to be associative, all we need to do to verify this is to find one instance of the lack of associativity. For the operation * defined on the set Thus

above, we find that

, while

.

, so the operation * is not associative

3.3 Semigroup: A mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. A semigroup is an associative groupoid. A semigroup with an identity is called a monoid. A semigroup can be empty. 3.4 Monoid: A monoid is a set that is closed under an associative binary operation and has an identity element such that for all , . Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element. A monoid must contain at least one element. A monoid that is commutative is, not surprisingly, known as a commutative monoid. 3.5 Group: A group G is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements A, B, C, ... with binary operation between A and B denoted AB form a group if 1. Closure: If A and B are two elements in G, then the product AB is also in G. 2. Associativity: The defined multiplication is associative, i.e., for all .

,

3. Identity: There is an identity element I (a.k.a. 1, , or ) such that element .

for every

4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element. Therefore, for each element A of G , the set contains an element such that . A group is a monoid each of whose elements is invertible.

A group must contain at least one element, with the unique single-element group known as the trivial group. The study of groups is known as group theory. If there are a finite number of elements, the group is called a finite group and the number of elements is called the group order of the group. A subset of a group that is closed under the group operation and the inverse operation is called a subgroup. Subgroups are also groups and many commonly encountered groups are in fact special subgroups of some more general larger group. A basic example of a finite group is the symmetric group Sn, which is the group of permutations (or "under permutation") of n objects. The simplest infinite group is the set of integers under usual addition. For continuous groups, one can consider the real numbers or the set of n X n invertible matrices. These last two are examples of Lie groups. 3.6 Groupoid:

There are at least two definitions of "groupoid" currently in use. The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set S returns a value which is itself a member of S ). Associativity, commutativity, etc., are not required. A groupoid can be empty. An associative groupoid is called a semigroup. The second type of groupoid is an algebraic structure first defined by Brandt (1926) and also known as a virtual group. A groupoid with base B (or "over B ") is a set Gwith mappings α and β from G onto B and a partially defined binary operation , satisfying the following four conditions: 1.

is defined whenever

2. Associativity: if either of 3. For each

, and in this case and

and

are defined so is the other and they are equal.

, there are left- and right-identity elements

and

. 4. Each

has an inverse

.

satisfying

Any group is a groupoid with base a single point.

and

.

respectively, satisfying

The most basic example of groupoid with base is the pair groupoid, where , and , , and with multiplication . Any equivalence relation on defines a subgroupoid of the pair groupoid. A useful way to think of a groupoid is as a parametrized equivalence relation on , as follows. Given a groupoid over , define an equivalence relation on by for each . This equivalence relation is "parameterized" because there may be more than one element in which give rise to the same equivalence, that is, and such that and .