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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 15 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 10947 times total.

Classification:
AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
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