Definition:Trichotomy

Definition

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:

$a \mathrel \RR b$

$a = b$

$b \mathrel \RR a$

Example

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

Also see

• Trichotomy Law: an ordering $\prec$ is a strict total ordering if and only if $\prec$ is a trichotomy.

• Results about trichotomies can be found here.