Description

• Title:
  Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
• Creator:
  Łazarz, Marcin
• Contributor:
  University of Wrocław, Department of Logic and Methodology of Sciences;
  lazarzmarcin@poczta.onet.pl
• Description:
  In the paper we investigate Birkhoff’s conditions (Bi) and (Bi*). We prove that a discrete lattice L satisfies the condition (Bi) (the condition (Bi*)) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S*7 (the lattice S7). As a corollary we obtain a well known result of J. Jakubík from [6]. Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S*7 ≪ L, in symbols. It is shown that an upper continuous lattice L satisfies (Bi*) if and only if L is a 4-cell lattice such that S7 ≪ L. The final corollary is a generalization of Jakubík’s theorem for upper continuous and strongly atomic lattices.
• Object availability:
  free access
• Rights:
  © Copyright by Authors, Łódź 2016; © Copyright for this edition by Uniwersytet Łódzki, Łódź 2016
• Date:
  2017.07.10;
  2016
• Type:
  Article
• Language:
  English
• Format:
  application/pdf
• Publisher:
  Wydawnictwo Uniwersytetu Łódzkiego
• Relation:
  Bulletin of the Section of Logic;3/4;
  45
• Subject:
  Birkhoff’s conditions;
  semimodularity conditions;
  modular lattice;
  discrete lattices;
  upper continuous lattice;
  strongly atomic lattice;
  cover-preserving sublattice;
  cell;
  4-cell lattice
• Identifier:
  0138-0680;
  http://hdl.handle.net/11089/22185 ;
  2449-836X;
  10.18778/0138-0680.45.3.4.04
• Data provider:
  Repozytorium Uniwersytetu Łódzkiego
• Can I use it?:
  seek permission
• Type:
  image