2. Dual ideals (filters) in a lattice $L$

(1) A dual ideal or filter in $L$ is a nonempty upset closed under meets.

(2) A dual ideal $D$ is principal if $D = [ a )$ for some $a \in D$.

(3) A dual ideal $D$ is proper if $D < L$.

(4) A dual ideal $D$ is prime if $D$ is proper and $x \vee y \in D$ implies $x \in D$ or $y \in D$.

(5) A dual ideal $D$ is maximal (an ultrafilter) if $D$ is proper and there is no dual ideal $E$ with $D < E < L$.

Note. Often the the terms filter and ultrafilter are reserved for the case where $L$ is a lattice of subsets, i.e., a sublattice of Pow$(X)$ for some $X$.