Category:Definitions/Orderings

This category contains definitions related to Orderings.
Related results can be found in Category:Orderings.

Let $S$ be a set.

$\RR$ 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 $

$\RR$ 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:

$\RR^{-1}$ denotes the inverse of $\RR$

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

Subcategories

This category has only the following subcategory.

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

\((1)\)	$:$	$\RR$ 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	\(\ds \forall a, b \in S:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \)

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