Axiom:Semigroup Axioms

Definition

A semigroup 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 $

These stipulations can be referred to as the semigroup axioms.

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