Editing Partially ordered set (section)

{{Short description|Mathematical set with an ordering}}
{{Stack|{{Binary relations}}}}
[[Image:Hasse diagram of powerset of 3.svg|right|thumb|upright=1.15|'''Fig. 1''' The [[Hasse diagram]] of the [[Power set|set of all subsets]] of a three-element set <math>\{x, y, z\},</math> ordered by [[set inclusion|inclusion]]. Sets connected by an upward path, like <math>\emptyset</math> and <math>\{x,y\}</math>, are comparable, while e.g. <math>\{x\}</math> and <math>\{y\}</math> are not.]]
In [[mathematics]], especially [[order theory]], a '''partial order''' on a [[Set (mathematics)|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 order]]s, in which every pair is comparable.

Formally, a partial order is a [[homogeneous binary relation]] that is [[Reflexive relation|reflexive]], [[antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]]. A '''partially ordered set''' ('''poset''' for short) is an [[ordered pair]] <math>P=(X,\leq)</math> consisting of a set <math>X</math> (called the ''ground set'' of <math>P</math>) and a partial order <math>\leq</math> on <math>X</math>. When the meaning is clear from context and there is no ambiguity about the partial order, the set <math>X</math> itself is sometimes called a poset.