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