Axiom:Ordered Semigroup Axioms

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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.