Axiom:Ordering Axioms/Formulation 2

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:

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

where:

$\circ$ denotes relation composition

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

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

These criteria are called the ordering axioms.

Also see

• Axiom:Ordering Axioms/Formulation 1 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.