# 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 $S$
$\FF \ne S$

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

• Results about filters can be found here.