Definition:Lattice Filter/Definition 1
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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 $S$: | \(\ds \forall x, y \in F:\) | \(\ds x \wedge y, x \vee y \in F \) | |||||
\((\text {LF 2})\) | $:$ | \(\ds \forall x \in F: \forall a \in S:\) | \(\ds x \vee a \in F \) |
Also see
Sources
- 1971: George A. Grätzer: Lattice Theory: Chapter $1$: First Concepts, $\S 3$: Some Algebraic Concepts