Axiom:Ordering Axioms/Formulation 1

Definition

Let $S$ be a set.

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

$\RR$ is an ordering if and only if $\RR$ satisifes the axioms:

\((1)\)	$:$		$\RR$ 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	\(\ds \forall a, b \in S:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \)

These criteria are called the ordering axioms on $S$.

Also see

• Axiom:Ordering Axioms/Formulation 2 for an alternative formulation of the ordering axioms on a set.
• Axiom:Ordering Axioms/Class Formulation for a formulation of the ordering axioms on a class.
• Definition:Ordering
• Equivalence of Definitions of Ordering

• Results about orderings can be found here.