Definition:Ordered Structure

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


 * $(1): \quad \left({S, \circ}\right)$ is an algebraic structure
 * $(2): \quad \left({S, \preceq}\right)$ is an ordered set
 * $(3): \quad \preceq$ is compatible with $\circ$.

There are various breeds of ordered structure the same way that there are for algebraic structures:

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

Totally Ordered Structure
As above, this has its various sub-breeds.

Also known as
In order to reduce confusion with the concept of an ordered set, an ordered structure is sometimes referred to as an ordered algebraic structure.

Also see

 * Ordered Set: this is also sometimes 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.


 * Ordered Ring, in which the definition is subtly different.