Definition:Partially Ordered Set

Definition
A poset (convenient abbreviation for partially ordered set) is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering.

The poset $\left({S, \preceq}\right)$ is said to be partially ordered by $\preceq$.

In general, a poset can also be a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is an ordering which may or may not be partial.

Alternative Name
Some sources call this an ordered set, and prefer not to use the term partial.

However, according to ProofWiki house style, the advantage of being able to specify a difference between the various types of ordering outweighs any possible perceived inaccuracy in terminology.

Thus, on ProofWiki, poset and partially ordered set are the names to be used.

Also see

 * Totally ordered set (toset)
 * Well-ordered set (woset)