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.

The term order structure is also sometimes encountered.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$