ordering relation
Let $S$ be a set. An ordering relation is a relation $\le $ on $S$ such that, for every $a,b,c\in S$:

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Either $a\le b$, or $b\le a$,

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If $a\le b$ and $b\le c$, then $a\le c$,

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If $a\le b$ and $b\le a$, then $a=b$.
Equivalently, an ordering relation is a relation $\le $ on $S$ which makes the pair $(S,\le )$ into a totally ordered set^{}. Warning: In some cases, an author may use the term “ordering relation” to mean a partial order^{} instead of a total order.
Given an ordering relation $\le $, one can define a relation $$ by: $$ if $a\le b$ and $a\ne b$. The opposite ordering is the relation $\ge $ given by: $a\ge b$ if $b\le a$, and the relation $>$ is defined analogously.
Title  ordering relation 
Canonical name  OrderingRelation 
Date of creation  20130322 11:52:04 
Last modified on  20130322 11:52:04 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  9 
Author  djao (24) 
Entry type  Definition 
Classification  msc 0300 
Classification  msc 8100 
Classification  msc 1800 
Classification  msc 17B37 
Classification  msc 18D10 
Classification  msc 18D35 
Classification  msc 16W30 
Related topic  TotalOrder 
Related topic  PartialOrder 
Related topic  Relation 
Defines  opposite ordering 