# Monoid

Monoid . In Algebra it is said of the algebraic structure formed by the pair <G, *> , such that G is a non-empty set and * is a binary operation ; then it is true that * is closed and associative and that the neutral element exists in G for * .

In the case that <G, *> is monoid and the operation is commutative, it is said to be a commutative or abelian monoid .

## 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 .

It is said that G with the operation * is a monoid .

## Examples

• The integersand the sum make up a monoid since the sum is closed and associative, 0 is neutral.
• The natural numbersand the product are also a monoid where 1 is the neutral of the multiplication .
• The stringand the concatenation form a monoid having the empty string by neutral.
• Let any algebraic group be <M, *>, it is also a monoid.