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ordering relation (Definition)

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: $a < b$ 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.




"ordering relation" is owned by djao.
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See Also: total order, partial order, relation

Also defines:  opposite ordering
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Cross-references: total order, partial order, mean, term, totally ordered set, relation
There are 17 references to this entry.

This is version 4 of ordering relation, born on 2001-10-21, modified 2004-03-20.
Object id is 444, canonical name is OrderingRelation.
Accessed 12956 times total.

Classification:
AMS MSC03-00 (Mathematical logic and foundations :: General reference works )

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