Geodesic
Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
Bulletin of the Section of Logic, Tome 45 (2016) no. 3-4.

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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.  
Keywords: Birkhoff’s conditions, semimodularity conditions, modular lattice, discrete lattices, upper continuous lattice, strongly atomic lattice, cover-preserving sublattice, cell, 4-cell lattice 
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     author = {{\L}azarz, Marcin},
     title = {Characterization of {Birkhoff{\textquoteright}s} {Conditions} by {Means} of {Cover-Preserving} and {Partially} {Cover-Preserving} {Sublattices}},
     journal = {Bulletin of the Section of Logic},
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     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BSL_2016_45_3-4_a0/}
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Łazarz, Marcin. Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices. Bulletin of the Section of Logic, Tome 45 (2016) no. 3-4. http://geodesic.mathdoc.fr/item/BSL_2016_45_3-4_a0/