Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
DOI:
https://doi.org/10.18778/0138-0680.45.3.4.04Keywords:
Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell latticeAbstract
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.
References
[1] G. Birkhoff, T.C. Bartee, Modern applied algebra, McGraw-Hill Book Company XII, New York etc. (1970).
Google Scholar
[2] P. Crawley, R.P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1973).
Google Scholar
[3] E. Fried, G. Gr¨atzer, H. Lakser, Projective geometries as cover-preserving sublattices, Algebra Universalis 27 (1990), pp. 270–278.
Google Scholar
[4] G. Grätzer, General lattice theory, Birkh¨auser, Basel, Stuttgart (1978).
Google Scholar
[5] G. Grätzer, E. Knapp, Notes on planar semimodular lattices. I: Construction, Acta Sci. Math. 73, No. 3–4 (2007), pp. 445–462.
Google Scholar
[6] J. Jakubík, Modular lattice of locally finite length, Acta Sci. Math. 37 (1975), pp. 79–82.
Google Scholar
[7] M. Łazarz, K. Siemieńczuk, Modularity for upper continuous and strongly atomic lattices Algebra Universalis 76 (2016), pp. 493–95.
Google Scholar
[8] S. MacLane, A conjecture of Ore on chains in partially ordered sets, Bull. Am. Math. Soc. 49 (1943), pp. 567–568.
Google Scholar
[9] O. Ore, Chains in partially ordered sets, Bull. Am. Math. Soc. 49 (1943), pp. 558–566.
Google Scholar
[10] M. Ramalho, On upper continuous and semimodular lattices, Algebra Universalis 32 (1994), pp. 330–340.
Google Scholar
[11] M. Stern, Semimodular Lattices. Theory and Applications, Cambridge University Press (1999).
Google Scholar
[12] A. Walendziak, Podstawy algebry ogólnej i teorii krat, Wydawnictwo Naukowe PWN, Warszawa (2009).
Google Scholar
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