Definition:Filter

Definition

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

A subset $\FF \subseteq S$ is called a filter of $\struct {S, \preccurlyeq}$ (or a filter on $\struct {S, \preccurlyeq}$) if and only if $\FF$ satisfies the filter axioms:

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

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

• Definition:Filter on Set

• Results about filters can be found here.