cover image

Partially ordered set

Mathematical set with an ordering / From Wikipedia, the free encyclopedia

Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Partial Order?

Summarize this article for a 10 years old


In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.

Fig.1 The Hasse diagram of the set of all subsets of a three-element set ordered by inclusion. Sets connected by an upward path, like and , are comparable, while e.g. and are not.

Formally, a partial order is a homogeneous binary relation that is reflexive, transitive and antisymmetric. A partially ordered set (poset for short) is a set on which a partial order is defined.