Partially ordered set: Difference between revisions

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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> 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.

== Partial order relations ==