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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 $\struct {S, \preccurlyeq}$.


$\FF$ 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 $S$:      \(\ds \forall x, y \in F:\) \(\ds x \wedge y \in F \)      


$F$ 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.