Definition:Ordered Structure

From ProofWiki
Jump to: navigation, search

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 Ring, in which the definition is subtly different.