Completely Representable Lattices

TL;DR

This paper explores the structure of bounded distributive lattices that can be represented with certain preservation properties, establishing strict class inclusions and their model-theoretic characteristics.

Contribution

It characterizes classes of representable lattices with preservation of joins and meets, and proves their strict inclusions and model-theoretic properties.

Findings

CRL is a strict subset of biCRL

biCRL is a strict subset of jCRL and mCRL

CRL is not elementary

Abstract

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be the intersection of jCRL and mCRL. We prove CRL is a strict subset of biCRL which is a strict subset of both jCRL and mCRL. Let L be a DL. Then L is in mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L is in jCRL iff L has a distinguishing set of completely prime filters, and L is in CRL iff L has a distinguishing set of complete, completely prime filters. Each of the classes above is shown to be pseudo-elementary and hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.