Definition:Ordered Semigroup

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An ordered semigroup is an ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a semigroup.

Ordered Semigroup Axioms

The properties that define an ordered semigroup can be gathered together as follows:

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

\((OS0)\)   $:$   Closure      \(\displaystyle \forall a, b \in S:\) \(\displaystyle a \circ b \in S \)             
\((OS1)\)   $:$   Associativity      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)             
\((OS2)\)   $:$   Compatibility of $\preceq$ with $\circ$      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} \)             
where $\preceq$ is an ordering    \(\displaystyle a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} \)             

Also see

  • Results about ordered semigroups can be found here.