# Category:Orderings

This category contains results about Orderings.
Definitions specific to this category can be found in Definitions/Orderings.

Let $S$ be a set.

### Definition 1

An ordering on $S$ is a relation $\RR$ on $S$ such that:

 $(1)$ $:$ $\RR$ 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

An ordering on $S$ is a relation $\RR$ on $S$ such that:

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

where:

$\circ$ denotes relation composition
$\RR^{-1}$ denotes the inverse of $\RR$
$\Delta_S$ denotes the diagonal relation on $S$.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Orderings"

The following 19 pages are in this category, out of 19 total.