Definition:Ordered Structure

Definition
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 an ordered set;
 * 3) $$\preceq$$ is compatible with $$\circ$$.

Ordered Structures
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.

Totally Ordered Structure
When the ordering in an ordered structure is a total ordering, the structure is then a totally ordered structure.

As above, this has its various sub-breeds.

Ordered Set
An ordered set can also be referred to as an ordered structure, or sometimes an order structure, on the grounds that it is a relational structure which happens to be an ordering.