Partially ordered set

A non-strict partial order is also known as an antisymmetric preorder.

One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig.4):

All three can similarly be defined for the Cartesian product of more than two sets.

The number of strict partial orders is the same as that of partial orders.

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders.