Filter Classes of Upsets of Distributive Lattices

Let us say that a class of upward closed sets (upsets) distributive lattices is finitary filter if it under homomorphic preimages, intersections, and directed unions. We show the only classes upsets are formed by what we call n-filters. These related to finite Boolean lattice with n atoms in same way filters two-element lattice: n-filters precisely intersections prime pre-images n-filter non-zero elements atoms. Moreover, on algebras generated upsets.