ordering relation

Let S be a set. An ordering relation is a relation on S such that, for every a,b,cS:

  • Either ab, or ba,

  • If ab and bc, then ac,

  • If ab and ba, then a=b.

Equivalently, an ordering relation is a relation on S which makes the pair (S,) into a totally ordered setMathworldPlanetmath. Warning: In some cases, an author may use the term “ordering relation” to mean a partial orderMathworldPlanetmath instead of a total order.

Given an ordering relation , one can define a relation < by: a<b if ab and ab. The opposite ordering is the relation given by: ab if ba, and the relation > is defined analogously.

Title ordering relation
Canonical name OrderingRelation
Date of creation 2013-03-22 11:52:04
Last modified on 2013-03-22 11:52:04
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 03-00
Classification msc 81-00
Classification msc 18-00
Classification msc 17B37
Classification msc 18D10
Classification msc 18D35
Classification msc 16W30
Related topic TotalOrder
Related topic PartialOrder
Related topic Relation
Defines opposite ordering