ordering relation (original) (raw)

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.