Completely representable lattices

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 ${{\bf jCRL} \cap {\bf mCRL}}$ . We prove

$${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$$

where the marked inclusions are proper.

Let L be a DL. Then ${L \in {\bf mCRL}}$ iff L has a distinguishing set of complete, prime filters. Similarly, ${L \in {\bf jCRL}}$ iff L has a distinguishing set of completely prime filters, and ${L \in {\bf CRL}}$ iff L has a distinguishing set of complete, completely prime filters.

Each of the classes above is shown to be pseudo-elementary, hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.