BINARY OPERATIONS, IDENTITY ELEMENTS, INVERSE ELEMENTS

Def. 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.

Def. 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.

Types of binary operations.

Commutative. A binary operation on a set S is called commutative if xy = yx for all x,y in S.

Associative. 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.

Def. 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.

Def. 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 ... x_n = x₁(x₂x₃)(x₄ ... x_n) = (x₁x₂)(x₃x₄)(x₅ ...x_n)

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.

References.

  Ayres. Modern Algebra. p. 19, 20

  Beaumont, Ball. Introduction to Modern Algebra and Matrix Theory. p. 123 - 127