# Category:Axioms/Preordering Axioms

This category contains axioms related to Preordering Axioms.

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

### Definition 1

$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the preordering axioms:

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

### Definition 2

$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the preordering axioms:

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

where:

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

## Pages in category "Axioms/Preordering Axioms"

The following 3 pages are in this category, out of 3 total.