Definition:Semigroup Axioms

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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      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)             

These stipulations can be referred to as the semigroup axioms.