Commutative semi-group

Commutative or Abelian semi-group . In Algebra it is said of the algebraic structure conformed by the pair <G, *> , such that G is a non-empty set and * is a binary operation ; then it is true that * is closed , associative and commutative .

Definition

Let be a set G and the binary operation * defined as * (x, y) = z , normally written as x * y = z that satisfies the axioms :

  1. Closing: . * is closed.
  2. Associativity: For all x , y , z of G , (x * y) * z = x * (y * z) .
  3. Commutativity: For all x , y of G , x * y = y * x .

It is said that G with the operation * is commutative or Abelian semigroup .

In other words, an abelian semigroup is a semigroup whose operation is also commutative.

Examples

  • Every commutative group <G, *>is a commutative semigroup.
  • The following are abelian semigroups represented in tabular form:
<{a, b}, @> <{a, b, c}, *>
@ to b
to to b
b b to
* to b c
to to b c
b b c to
c c to b
  • It is a commutative semi-group of complex matrices with the operation of addition of matrices defined in the traditional way:
    • with and

 

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