Definition:Ordered Set

Definition
An ordered set is a relational structure $\left({S, \preceq}\right)$ 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.

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.

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