Definition:Monoid Axioms

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

\((\text S 0)\)   $:$   Closure      \(\displaystyle \forall a, b \in S:\) \(\displaystyle a \circ b \in S \)             
\((\text S 1)\)   $:$   Associativity      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)             
\((\text S 2)\)   $:$   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.

Also see