# ordering relation

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

• Either $a\leq b$, or $b\leq a$,

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

• If $a\leq b$ and $b\leq a$, then $a=b$.

Equivalently, an ordering relation is a relation $\leq$ on $S$ which makes the pair $(S,\leq)$ 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 $\leq$, one can define a relation $<$ by: $a if $a\leq b$ and $a\neq b$. The opposite ordering is the relation $\geq$ given by: $a\geq b$ if $b\leq a$, 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