Category:Preorderings

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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 $S$.