# Category:Axioms/Ordering Axioms

This category contains axioms related to Ordering Axioms.

Let $S$ be a set.

### Definition 1

$\RR$ is an ordering on $S$ if and only if $\RR$ satisfies the ordering 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 $\ds \forall a, b \in S:$ $\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b$

### Definition 2

$\RR$ is an ordering on $S$ if and only if $\RR$ satisfies the ordering 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 $S$.

## Pages in category "Axioms/Ordering Axioms"

The following 4 pages are in this category, out of 4 total.