Definition:Lattice Filter

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

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

Also known as
In some sources a (lattice) filter is called a dual ideal.

Also see

 * Equivalence of Definitions of Lattice Filter, where it is shown that the alternative definitions of a filter are equivalent.


 * Definition:Filter (Meet Semilattice), where a filter is defined in the context of a meet semilattice


 * Equivalence of Definitions of Lattice Filter, where it is shown that in a lattice the definition of (lattice) filter is equivalent to the definition of (meet semilattice) filter.


 * Definition:Filter, where a filter is defined in the more general context of Order Theory


 * Meet Semilattice Filter iff Ordered Set Filter, where it is shown that in a meet semilattice the definition of (meet semilattice) filter is equivalent to the definition of (ordered set) filter.


 * Definition:Ideal (Lattice), where the concept of ideal, the dual concept of a filter, is defined.