Definition:Partially Ordered Set

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

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

Also known as
The word poset is frequently to be found in the literature, but this is frequently understood to mean a general ordered set which may be either partial or total.

Some sources use the term partly ordered set.

Also see

 * Definition:Ordered Set
 * Definition:Totally Ordered Set
 * Definition:Well-Ordered Set


 * Definition:Strictly Ordered Set
 * Definition:Strictly Partially Ordered Set
 * Definition:Strictly Totally Ordered Set
 * Definition:Strictly Well-Ordered Set