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 $\RR$ satisifes the axioms:

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