partial order

A partial order (often simply referred to as an order or ordering) is a relation $\leq\>\subset A\times A$ that satisfies the following three properties:

1. 1.

Reflexivity: $a\leq a$ for all $a\in A$

2. 2.

Antisymmetry: If $a\leq b$ and $b\leq a$ for any $a,b\in A$, then $a=b$

3. 3.

Transitivity: If $a\leq b$ and $b\leq c$ for any $a,b,c\in A$, then $a\leq c$

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

• Comparability: For any $a,b\in A$, either $a\leq b$ or $b\leq a$.

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 $\leq$ is defined as a transitive irreflexive binary relation (on a set). As a result, if $a\leq b$, then $b\nleq a$, and therefore $\leq$ 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 QuasiOrder Related topic CategoryAssociatedToAPartialOrder Related topic OrderingRelation Related topic HasseDiagram Related topic NetsAndClosuresOfSubspaces