nLab trichotomous relation

 Definition

A trichotomous relation on a set $S$ is a relation $R(x, y)$ which satisfies trichotomy:

• for all $x \in S$ and $y \in S$, exactly one of $R(x, y)$, $R(y, x)$, $x = y$ is true.

 Examples

• In classical mathematics, the law of trichotomy or axiom of trichotomy for the real numbers state that the strict total order on the real numbers is trichotomous. In constructive mathematics, this statement is equivalent to the analytic LPO.

• In classical mathematics, every strict total order is a trichotomous relation, and strict total orders can be defined as a transitive trichotomous relation. However, in constructive mathematics, only the decidable strict total orders are trichotomous relations.

 Related concepts

• strict total order

• connected relation

• analytic LPO