Definition:Filter

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Definition

Let $\struct {S, \preccurlyeq}$ be an ordered set.


A filter of $\struct {S, \preccurlyeq}$ (or a filter on $\struct {S, \preccurlyeq}$) is a subset $\FF \subseteq S$ which satisfies the following conditions:

\((1)\)   $:$   \(\ds \FF \ne \O \)             
\((2)\)   $:$   \(\ds x, y \in \FF \implies \exists z \in \FF: z \preccurlyeq x, z \preccurlyeq y \)             
\((3)\)   $:$   \(\ds \forall x \in \FF: \forall y \in S: x \preccurlyeq y \implies y \in \FF \)             


Proper Filter

Let $\FF$ be a filter on $\struct {S, \preccurlyeq}$.


Then:

$\FF$ is a proper filter on $S$

if and only if:

$\FF \ne S$

That is, if and only if $\FF$ is a proper subset of $S$.


Also see

  • Results about filters can be found here.