Axiom:Preordering Axioms/Formulation 2

Definition

Let $\RR \subseteq S \times S$ be a relation on a set $S$.

$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the axioms:

\((1)\)	$:$		$\RR$ is transitive		\(\ds \RR \circ \RR = \RR \)
\((2)\)	$:$		$\RR$ is reflexive		\(\ds \Delta_S \subseteq \RR \)

where:

$\circ$ denotes relation composition

$\Delta_S$ denotes the diagonal relation on $S$.

These criteria are called the preordering axioms.

Also see

• Axiom:Preordering Axioms/Formulation 1 for an alternative formulation for the preordering axioms