##### Wiktionary

**partially ordered set**

n. 1 (context set theory English) A set having a specified partial order. 2 (context set theory English) Said set together with said partial order; the ordered pair of said set and said partial order.

##### Wikipedia

**Partially ordered set**

In mathematics, especially order theory, a **partially ordered set** (or **poset**) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a *partial order* to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation.

A familiar real-life 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 bear no such relationship.