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 $S$.

Meet Semilattice

Let $\struct {S, \wedge, \preccurlyeq}$ be a meet semilattice.

Let $F \subseteq S$ be a non-empty subset of $S$.

Then $F$ is a meet semilattice ideal of $S$ if and only if $F$ satisifies the meet semilattice filter axioms:

\((\text {MSF 1})\)	$:$		$F$ is an upper section of $S$:	\(\ds \forall x \in F: \forall y \in S:\)	\(\ds x \preccurlyeq y \implies y \in F \)
\((\text {MSF 2})\)	$:$		$F$ is a subsemilattice of $S$:	\(\ds \forall x, y \in F:\)	\(\ds x \wedge y \in F \)

Lattice

$F$ is a lattice filter of $S$ if and only if $F$ satisifes the lattice filter axioms:

\((\text {LF 1})\)	$:$		$F$ is a sublattice of $S$:	\(\ds \forall x, y \in F:\)	\(\ds x \wedge y, x \vee y \in F \)
\((\text {LF 2})\)	$:$			\(\ds \forall x \in F: \forall a \in S:\)	\(\ds x \vee a \in F \)

Also see

• Definition:Filter on Set

• Meet Semilattice Filter iff Ordered Set Filter
• Equivalence of Definitions of Lattice Filter

• Definition:Ideal (Order Theory)

• Results about filters can be found here.

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {VII}$: Continuous Lattices, Definition $2.5$