Completely representable lattices

作者：Egrot Robert^*; Hirsch Robin

来源：Algebra Universalis, 2012, 67(3): 205-217.

DOI：10.1007/s00012-012-0181-4

• distributive lattice
• complete representation

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 jCRLn mCRL. We prove CRL subset of biCRL = mCRL boolean AND jCRL subset of mCRL not equal jCRL subset of DL where the marked inclusions are proper. Let L be a DL. Then L. mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L. jCRL iff L has a distinguishing set of completely prime filters, and L. 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.

• 出版日期2012-5