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

 * User:Leigh.Samphier/OrderTheory/Definition:Filter (Lattice)/Definition 2

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