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