Definition:Trichotomy

Definition
Let $$S$$ be a set.

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


 * $$a \mathcal{R} b$$;
 * $$a = b$$;
 * $$b \mathcal{R} a$$.

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

From the Trichotomy Law, we have that a poset $$\left({S; \preceq}\right)$$is a toset iff $$\prec$$ is also a trichotomy.