Definition:Lattice Filter
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Definition
Let $\struct {S, \vee, \wedge, \preccurlyeq}$ be a lattice.
Let $F \subseteq S$ be a non-empty subset of $S$.
Definition 1
$F$ is a lattice filter of $S$ if and only if $F$ satisifes the lattice filter axioms:
\((\text {LF 1})\) | $:$ | $F$ is a sublattice of $S$: | \(\ds \forall x, y \in F:\) | \(\ds x \wedge y, x \vee y \in F \) | |||||
\((\text {LF 2})\) | $:$ | \(\ds \forall x \in F: \forall a \in S:\) | \(\ds x \vee a \in F \) |
Definition 2
$F$ is a lattice filter of $S$ if and only if $F$ is a meet semilattice filter.
Also known as
In some sources a lattice filter is called a dual ideal.
Also see
- Definition:Meet Semilattice Filter
- Definition:Filter
- Meet Semilattice Filter iff Ordered Set Filter
- Definition:Lattice Ideal
- Results about lattice filters can be found here.