# Monoid

From Maths

Not to be confused with group

## Definition

A **monoid**^{[1]} is a set [ilmath]S[/ilmath] and a function [ilmath]\times_S:S\times S\rightarrow S[/ilmath] (called the operation) such that [ilmath]\times_S[/ilmath] is:

- Associative - that is [math]\forall x,y,z\in S[(xy)z=x(yz)][/math]
- Has identity element - that is [math]\exists e\in S\forall x\in S[ex=xe=x][/math]

(Here [ilmath]xy[/ilmath] denotes [ilmath]\times_S(x,y)[/ilmath] which being an operator would be written [ilmath]x\times_S y[/ilmath])

### Abelian monoid

A monoid is **Abelian** or **commutative** if:

- [ilmath]\forall x,y\in S[xy=yx][/ilmath]

## See also

## References

- ↑ Algebra - Serge Lang - Revised Third Edition - Graduate Texts In Mathematics