Filter classes of upsets of distributive lattices

TL;DR

This paper characterizes all finitary filter classes of upward closed sets in distributive lattices, showing they are exactly the $n$-filters related to finite Boolean lattices with $n$ atoms.

Contribution

It establishes that the only finitary filter classes are the $n$-filters, connecting them to prime filters and homomorphic preimages of finite Boolean lattices.

Findings

Finitary filter classes are exactly the $n$-filters.

$n$-filters are intersections of prime $n$-filters.

On Boolean algebras, these classes are generated by prime upsets.

Abstract

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