Definition:Lattice Filter

Definition

Let $\struct {S, \vee, \wedge, \preccurlyeq}$ be a lattice.

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

$F$ 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 $

$F$ is a lattice filter of $S$ if and only if $F$ is a meet semilattice filter.

In some sources a lattice filter is called a dual ideal.

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