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

• Autorzy: Marcin Łazarz
• Języki publikacji: EN

Abstrakty

EN

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.  

Słowa kluczowe

EN

Birkhoff’s conditions; semimodularity conditions; modular lattice; discrete lattices; upper continuous lattice; strongly atomic lattice; cover-preserving sublattice; cell; 4-cell lattice

• Wydawca: Wydawnictwo Uniwersytetu Łódzkiego
• Czasopismo: Bulletin of the Section of Logic
• Rocznik: 2016
• Tom: 45
• Numer: 3/4

Daty

wydano

2016-12-30

Twórcy

autor

Marcin Łazarz

• University of Wrocław, Department of Logic and Methodology of Sciences

Bibliografia

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Identyfikatory

DOI

10.18778/0138-0680.45.3.4.04