partial order

A partial orderMathworldPlanetmath (often simply referred to as an order or ordering) is a relationMathworldPlanetmath A×A that satisfies the following three properties:

  1. 1.

    ReflexivityMathworldPlanetmath: aa for all aA

  2. 2.

    Antisymmetry: If ab and ba for any a,bA, then a=b

  3. 3.

    Transitivity: If ab and bc for any a,b,cA, then ac

A total orderMathworldPlanetmath is a partial order that satisfies a fourth property known as comparability:

  • Comparability: For any a,bA, either ab or ba.

A set and a partial order on that set define a poset.

Remark. In some literature, especially those dealing with the foundations of mathematics, a partial order is defined as a transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath irreflexiveMathworldPlanetmath binary relation (on a set). As a result, if ab, then ba, and therefore is antisymmetric.

Title partial order
Canonical name PartialOrder
Date of creation 2013-03-22 11:43:32
Last modified on 2013-03-22 11:43:32
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 24
Author mathcam (2727)
Entry type Definition
Classification msc 06A06
Classification msc 35C10
Classification msc 35C15
Classification msc 55-01
Classification msc 55-00
Synonym order
Synonym partial ordering
Synonym ordering
Related topic Relation
Related topic TotalOrder
Related topic Poset
Related topic BinarySearch
Related topic SortingProblem
Related topic ChainCondition
Related topic PartialOrderWithChainConditionDoesNotCollapseCardinals
Related topic QuasiOrderMathworldPlanetmath
Related topic CategoryAssociatedToAPartialOrder
Related topic OrderingRelation
Related topic HasseDiagram
Related topic NetsAndClosuresOfSubspaces