Definition:Semigroup Axioms

A semigroup is an algebraic structure $\struct {S, \circ}$ 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 $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c$