# Axiom:Ordered Semigroup Axioms

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## Definition

An ordered semigroup is an algebraic system $\struct {S, \circ, \preceq}$ which satisfies the following properties:

 $(\text {OS} 0)$ $:$ Closure $\ds \forall a, b \in S:$ $\ds a \circ b \in S$ $(\text {OS} 1)$ $:$ Associativity $\ds \forall a, b, c \in S:$ $\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c$ $(\text {OS} 2)$ $:$ Compatibility of $\preceq$ with $\circ$ $\ds \forall a, b, c \in S:$ $\ds a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c}$ where $\preceq$ is an ordering $\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b}$

These stipulations can be referred to as the ordered semigroup axioms.