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 Semigroup

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

Ordered Commutative Semigroup

An ordered commutative semigroup is an ordered semigroup $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a commutative 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 $\preceq$ is a total ordering, the structure $\left({S, \circ, \preceq}\right)$ is then a 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.