eprintid: 1056690 rev_number: 36 eprint_status: archive userid: 608 dir: disk0/01/05/66/90 datestamp: 2011-02-14 21:19:32 lastmod: 2022-01-01 23:08:05 status_changed: 2011-02-14 21:19:32 type: article metadata_visibility: show item_issues_count: 0 creators_name: Egrot, R creators_name: Hirsch, R title: Completely representable lattices ispublished: pub divisions: UCL divisions: B04 divisions: C05 divisions: F48 abstract: t 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. date: 2012 official_url: http://dx.doi.org/10.1007/s00012-012-0181-4 vfaculties: VENG oa_status: green language: eng primo: open primo_central: open_green verified: verified_manual elements_source: Scopus elements_id: 290121 doi: 10.1007/s00012-012-0181-4 lyricists_name: Hirsch, Robin lyricists_id: RHIRS08 full_text_status: public publication: Algebra Universalis volume: 67 number: 3 pagerange: 205 - 217 issn: 0002-5240 citation: Egrot, R; Hirsch, R; (2012) Completely representable lattices. Algebra Universalis , 67 (3) 205 - 217. 10.1007/s00012-012-0181-4 <https://doi.org/10.1007/s00012-012-0181-4>. Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/1056690/1/AUCRL.pdf