Axiom:Preordering Axioms

Definition

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

$\RR$ satisfies the preordering axioms on $S$ if and only if:

$(1)$	$:$		$\RR$ is reflexive	$\ds \forall a \in S:$	$\ds a \mathrel \RR a $
$(2)$	$:$		$\RR$ is transitive	$\ds \forall a, b, c \in S:$	$\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c $

$\RR$ satisfies the preordering axioms on $S$ if and only if:

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

where:

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

\((1)\)	$:$		$\RR$ is reflexive	\(\ds \forall a \in S:\)	\(\ds a \mathrel \RR a \)
\((2)\)	$:$		$\RR$ is transitive	\(\ds \forall a, b, c \in S:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)