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	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