Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices

Marcin Łazarz

Bulletin of the Section of Logic (2016)

  • Volume: 45, Issue: 3/4
  • ISSN: 0138-0680

Abstract

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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. Jakub´ık 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 S7 ≪ 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. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice.  

How to cite

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Marcin Łazarz. "Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices." Bulletin of the Section of Logic 45.3/4 (2016): null. <http://eudml.org/doc/295578>.

@article{MarcinŁazarz2016,
abstract = {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. Jakub´ık 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 S7 ≪ 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. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice.  },
author = {Marcin Łazarz},
journal = {Bulletin of the Section of Logic},
keywords = {Birkhoff’s conditions; semimodularity conditions; modular lattice; discrete lattices; upper continuous lattice; strongly atomic lattice; cover-preserving sublattice; cell; 4-cell lattice},
language = {eng},
number = {3/4},
pages = {null},
title = {Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices},
url = {http://eudml.org/doc/295578},
volume = {45},
year = {2016},
}

TY - JOUR
AU - Marcin Łazarz
TI - Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
JO - Bulletin of the Section of Logic
PY - 2016
VL - 45
IS - 3/4
SP - null
AB - 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. Jakub´ık 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 S7 ≪ 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. Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice.  
LA - eng
KW - Birkhoff’s conditions; semimodularity conditions; modular lattice; discrete lattices; upper continuous lattice; strongly atomic lattice; cover-preserving sublattice; cell; 4-cell lattice
UR - http://eudml.org/doc/295578
ER -

References

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  1. [1] G. Birkhoff, T.C. Bartee, Modern applied algebra, McGraw-Hill Book Company XII, New York etc. (1970). 
  2. [2] P. Crawley, R.P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1973). 
  3. [3] E. Fried, G. Gr¨atzer, H. Lakser, Projective geometries as cover-preserving sublattices, Algebra Universalis 27 (1990), pp. 270–278. 
  4. [4] G. Grätzer, General lattice theory, Birkh¨auser, Basel, Stuttgart (1978). 
  5. [5] G. Grätzer, E. Knapp, Notes on planar semimodular lattices. I: Construction, Acta Sci. Math. 73, No. 3–4 (2007), pp. 445–462. 
  6. [6] J. Jakubík, Modular lattice of locally finite length, Acta Sci. Math. 37 (1975), pp. 79–82. 
  7. [7] M. Łazarz, K. Siemieńczuk, Modularity for upper continuous and strongly atomic lattices Algebra Universalis 76 (2016), pp. 493–95. 
  8. [8] S. MacLane, A conjecture of Ore on chains in partially ordered sets, Bull. Am. Math. Soc. 49 (1943), pp. 567–568. 
  9. [9] O. Ore, Chains in partially ordered sets, Bull. Am. Math. Soc. 49 (1943), pp. 558–566. 
  10. [10] M. Ramalho, On upper continuous and semimodular lattices, Algebra Universalis 32 (1994), pp. 330–340. 
  11. [11] M. Stern, Semimodular Lattices. Theory and Applications, Cambridge University Press (1999). 
  12. [12] A. Walendziak, Podstawy algebry ogólnej i teorii krat, Wydawnictwo Naukowe PWN, Warszawa (2009). 

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