Definition:Ordered Structure

An ordered structure $$\left({S, \circ; \preceq}\right)$$ is a set $$S$$ such that:


 * 1) $$\left({S, \circ}\right)$$ is an algebraic structure;
 * 2) $$\left({S; \preceq}\right)$$ is a poset;
 * 3) $$\preceq$$ is compatible with $$\circ$$.

Of course, there are various breeds of ordered structure the same way that there are for algebraic structures.

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

Ordered Subsemigroup
An ordered subsemigroup $$\left({T, \circ; \preceq}\right)$$ of an ordered structure $$\left({S, \circ; \preceq}\right)$$ is an ordered semigroup such that the semigroup $$\left({T, \circ}\right)$$ is a subsemigroup of $$\left({S, \circ}\right)$$.

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

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

Ordered Subgroup
An ordered subgroup $$\left({T, \circ; \preceq}\right)$$ of an ordered structure $$\left({S, \circ; \preceq}\right)$$ is an ordered group such that the group $$\left({T, \circ}\right)$$ is a subgroup of $$\left({S, \circ}\right)$$.

The list goes on; we won't labour the point.