Monoid

Not to be confused with group

Definition

A monoid is a set $S$ and a function $\times_S:S\times S\rightarrow S$ (called the operation) such that $\times_S$ is:

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

(Here $xy$ denotes $\times_S(x,y)$ which being an operator would be written $x\times_S y$)

Abelian monoid

A monoid is Abelian or commutative if:

• $\forall x,y\in S[xy=yx]$

See also

• Group