An extension of Birkhoff's representation theorem to locally-finite distributive lattices

TL;DR

This paper extends Birkhoff's representation theorem to locally finite distributive lattices, providing a new isomorphism with order ideals of prime filters with finite symmetric difference from a fixed ideal.

Contribution

It introduces a novel representation theorem for locally finite distributive lattices based on prime filters and finite symmetric differences, expanding classical lattice theory.

Findings

New representation theorem for locally finite distributive lattices

Isomorphism with order ideals of prime filters with finite symmetric difference

Simplified extension of Stone's theorem to general distributive lattices

Abstract

Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. We present a simplified version of Stone's extension of this theorem to general distributive lattices. We then apply this formulation to locally finite distributive lattices to produce a novel representation theorem: The lattice is isomorphic to the order ideals of the poset of prime filters of the lattice whose symmetric difference from a particular ideal is finite.