Category:Axioms/Preordering Axioms

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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.