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Let $S$ be a set.

A trichotomy on $S$ is a relation $\RR$ on $S$ such that for every pair of elements $a, b \in S$, exactly one of the following three conditions applies:

$a \mathrel \RR b$
$a = b$
$b \mathrel \RR a$


A classic example of a trichotomy is the standard less than ordering on the set of real numbers.

Also see

  • Results about trichotomies can be found here.