Definition:Ordered Set

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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:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation.

Let $A$ be a subclass of the field of $\RR$.

Let the restriction of $\RR$ to $A$ be an ordering on $A$.

Then $A$ is described as being ordered under $\RR$.

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ as a totally ordered set.

Also see

  • Results about ordered sets can be found here.