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.