  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.
