Axiom:Ordered Semigroup Axioms

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.

\((\text {OS} 0)\)	$:$	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		\(\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} \)