Axiom:Lattice Filter Axioms

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 axioms:

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

These criteria are called the lattice filter axioms.