Axiom:Monoid Axioms

Definition

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

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

\((\text S 0)\)	$:$	Closure	\(\ds \forall a, b \in S:\)	\(\ds a \circ b \in S \)
\((\text S 1)\)	$:$	Associativity	\(\ds \forall a, b, c \in S:\)	\(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)
\((\text S 2)\)	$:$	Identity	\(\ds \exists e_S \in S: \forall a \in S:\)	\(\ds e_S \circ a = a = a \circ e_S \)