Definition:Lattice Filter/Definition 1

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 $S$ $F$ satisifes the lattice filter axioms:

Also see

 * Definition:Filter (Lattice)/Definition 2, an alternative definition of a filter.


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