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Internal Binary Operations

 
 

Internal Binary Operation

An #~{internal binary operation} on a set ~A, is a function @F #: ~A # ~A -> ~A.
Write _ ~x &dot. ~y _ for _ @F ( ~x , ~y ). The set ~A with this operation [ denoted ( ~A, &dot. ) ] is called a #~{groupoid}.

Associativity

The operation &dot. is said to be #~{associative} _ <=> _ ( ~x &dot. ~y ) &dot. ~z _ = _ ~x &dot. ( ~y &dot. ~z )

Identity Element

The groupoid _ ( ~A , &dot. ) _ has an #~{identity element} , ~e , if

~e &dot. ~x _ = _ ~x &dot. ~e _ = _ ~x , _ for any ~x &in. ~A

The identity element, ~e , if it exists, is unique, for if also _ ~x &dot. ~e' _ = _ ~x , _ for all ~x &in. ~A, _ then _ ~e &dot. ~e' _ = _ ~e , _ but from the definition of ~e we have _ ~e &dot. ~e' _ = _ ~e' , _ so _ ~e _ = _ ~e' .

Inverse Of An Element

If ( ~A , &dot. ) has an identity element , ~e , _ ~x &in. ~A is said to have an ~#{inverse} , _ ~x' , _ if

~x &dot. ~x' _ = _ ~x' &dot. ~x _ = _ ~e

If ( ~A , &dot. ) is associative, then any element that has an inverse has a unique inverse, _ for if also _ ~x &dot. ~x" _ = _ ~e , _ then

~x' _ = _ ~x' &dot. ~e _ = _ ~x' &dot. ( ~x &dot. ~x" ) _ = _ ( ~x' &dot. ~x ) &dot. ~x" _ = _ ~e &dot. ~x" _ = _ ~x"

If ~x' is the inverse of ~x , _ then obviously ~x is the inverse of ~x' . I.e. _ ( ~x' )' _ = _ ~x .

Structures With One Internal Binary Operation

Semigroup

A #~{semigroup} is a groupoid with an associative binary operation.

Group

A #~{group} is a semigroup with an identity element, and in which every element has an inverse.

Abelian Group

An #~{abelian group} or #~{commutative group} is one in which _ ~x &dot. ~y _ = _ ~y &dot. ~x , _ for any ~x, ~y &in. ~A .

Multiplicative And Additive Notation

Examples of binary operations are multiplication and addition, the following notation is adopted:

Multiplication , _ ( ~G , # )

  • ~x &dot. ~y _ is written _ ~x # ~y _ or just _ ~x ~y
  • The identity element is _ 1 . _ So _ 1 ~x _ = _ ~x 1 _ = _ ~x .
  • The inverse of ~x is written as _ ~x^{-1} . _ So _ ~x ~x^{-1} _ = _ ~x^{-1} ~x _ = _ 1 .

Addition , _ ( ~G , + )

  • ~x &dot. ~y _ is written _ ~x + ~y
  • The identity element is _ 0 . _ So _ 0 + ~x _ = _ ~x + 0 _ = _ ~x .
  • The inverse of ~x is written as _ -~x . _ So _ ~x + -~x _ = _ -~x + ~x _ = _ 0 .

The difference between these two operations is only apparent in structures with both, i.e. .

It is usual to use the multiplicative notation in general, and the additional notation for a commutative operation, e.g. in an abelian group. Also, with respect to addition, it is usual to write _ ~x - ~y _ for _ ~x + -~y .