# Category:Preorderings

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

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:

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

$\mathcal R$ is a preordering on $S$ if and only if:

$(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
$(2): \quad \Delta_S \subseteq \mathcal R$

where:

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

## Subcategories

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

## Pages in category "Preorderings"

The following 10 pages are in this category, out of 10 total.