ordering relation

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

• Either $a\le b$, or $b\le a$,

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

• 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	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