# Definition:Monoid Axioms

## Definition

A monoid is an algebraic structure $\struct {S, \circ, e_S}$ which satisfies the following properties:

 $(S0)$ $:$ Closure $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b \in S$ $(S1)$ $:$ Associativity $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c$ $(S2)$ $:$ Identity $\displaystyle \exists e_S \in S: \forall a \in S:$ $\displaystyle e_S \circ a = a = a \circ e_S$

The element $e_S$ is called the identity element.

These stipulations can be referred to as the monoid axioms.