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

1.
Reflexivity^{}: $a\le a$ for all $a\in A$

2.
Antisymmetry: If $a\le b$ and $b\le a$ for any $a,b\in A$, then $a=b$

3.
Transitivity: If $a\le b$ and $b\le c$ for any $a,b,c\in A$, then $a\le c$
A total order^{} is a partial order that satisfies a fourth property known as comparability:

•
Comparability: For any $a,b\in A$, either $a\le b$ or $b\le a$.
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 $\le $ is defined as a transitive^{} irreflexive^{} binary relation (on a set). As a result, if $a\le b$, then $b\nleqq a$, and therefore $\le $ is antisymmetric.
Title  partial order 
Canonical name  PartialOrder 
Date of creation  20130322 11:43:32 
Last modified on  20130322 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 5501 
Classification  msc 5500 
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 