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.

Some sources call this an ordered set, and prefer not to use the term partial. However, it is the opinion of the author of this page that the convenience of being able to specify a difference between the various types of ordering outweighs any possible perceived inaccuracy in terminology.

Also see

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