ordering relation
Let S be a set. An ordering relation is a relation ≤ on S such that, for every a,b,c∈S:
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Either a≤b, or b≤a,
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If a≤b and b≤c, then a≤c,
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If a≤b and b≤a, then a=b.
Equivalently, an ordering relation is a relation ≤ on S which makes the pair (S,≤) 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 ≤, one can define a relation < by: a<b if a≤b and a≠b. The opposite ordering is the relation ≥ given by: a≥b if b≤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 |