# Definition:Ordered Semigroup

## Definition

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.