Axiom:Ordering Axioms

Definition

Let $S$ be a set.

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

Formulation 1

$\RR$ satisfies the ordering axioms if and only if:

\((1)\)	$:$		$\RR$ is reflexive	\(\ds \forall a \in S:\)	\(\ds a \mathrel \RR a \)
\((2)\)	$:$		$\RR$ is transitive	\(\ds \forall a, b, c \in S:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)
\((3)\)	$:$		$\RR$ is antisymmetric	\(\ds \forall a, b \in S:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \)

Formulation 2

$\RR$ satisfies the ordering axioms if and only if:

\((1)\)	$:$				\(\ds \RR \circ \RR \)
\((2)\)	$:$				\(\ds \RR \cap \RR^{-1} = \Delta_S \)

where:

$\circ$ denotes relation composition

$\RR^{-1}$ denotes the inverse of $\RR$

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

Class Theory

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation on $V$.

$\RR$ satisfies the ordering axioms if and only if:

\((1)\)	$:$		$\RR$ is reflexive	\(\ds \forall a \in \Field \RR:\)	\(\ds a \mathrel \RR a \)
\((2)\)	$:$		$\RR$ is transitive	\(\ds \forall a, b, c \in \Field \RR:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)
\((3)\)	$:$		$\RR$ is antisymmetric	\(\ds \forall a, b \in \Field \RR:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \)

where $\Field \RR$ denotes the field of $\RR$.

Also see

• Equivalence of Definitions of Ordering

• Results about orderings can be found here.