trichotomous relation

A trichotomous relation is a homogeneous relation that is irreflexive, antisymmetric, and connected.

Formally,
a binary relation $R$ over a set $S$ is a trichotomous relation such that exactly one of

1. $xRy$,
2. $yRx$,
3. $x=y$
   holds for any two elements in the set.

A relation is a trichotomous relation if and only if it is asymmetric and connected.

Since the real numbers are a linearly ordered group for addition, the ordering is trichotomous; formally, for any $x,y\in \mathbb{R}$, exactly one of $x<y$, $y<x$, or $x=y$ holds. This is a generalization of the law of trichotomy for real numbers.