Definition:Ordered Set

Definition
An ordered set is a relational structure $\struct {S, \preceq}$ such that the relation $\preceq$ is an ordering.

Such a structure may be:


 * A partially ordered set (poset)
 * A totally ordered set (toset)
 * A well-ordered set (woset)

depending on whether the ordering $\preceq$ is:


 * A partial ordering
 * A total ordering
 * A well-ordering.

Ordered Class
The concept carries naturally over into class theory:

Also known as
Some sources call this an ordered structure, but this often has a more specialized meaning.

Some call this a poset, a partially ordered set or a partly ordered set, but we tend to avoid these on.

The term order structure is also sometimes encountered.

Some sources refer to $\struct {S, \preceq}$ as a partial order, calling $\preceq$ a partial order relation.

Also defined as
Some sources reserve the term ordered set for what is known on as a totally ordered set.