# Commutative Monoid

Commutative or abelian monoid . In Algebra it is said of the algebraic structure formed by the pair <G, *> , such that it is a monoid and * is commutative .

## Definition

Let be a set G and the binary operation * defined as * (x, y) = z or better x * y = z and each of the following axioms is satisfied :

1. Closing: . * is closed.
2. Associativity: For all x , y , z of G , (x * y) * z = x * (y * z) .
3. Existence of neutral: There is one and only one element e of G such that for all x of G it is true that x * e = e * x = x . and it is called I neutral for * in G .
4. Commutativity: For all x and y of G , it is true that x * y = y * x .

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

In other words:

• <G, *>is monoide commutative if and only if <G, *> is monoid and * is commutative for all elements of G .

## Examples

• The integersand the sum make up an abelian monoid since the sum is closed and associative, 0 is neutral.
• Natural numbersand product are also a commutative monoid where 1 is the neutral of multiplication .
• The naturals and the sum do not form a commutative monoid because although the sum is symmetric, the axiom of the existence of the neutral that invalidates the structure as a monoid is violated.
• The stringand the concatenation form a commutative monoid having the empty string by neutral.
• Let any algebraic group be <M, *>, it is also a monoid.