- Tytuł:
- Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
- Autorzy:
- Łazarz, Marcin
- Powiązania:
- https://bibliotekanauki.pl/articles/749906.pdf
- Data publikacji:
- 2016
- Wydawca:
- Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
- Tematy:
-
Birkhoff’s conditions
semimodularity conditions
modular lattice
discrete lattices
upper continuous lattice
strongly atomic lattice
cover-preserving sublattice
cell
4-cell lattice - Opis:
- 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.
- Źródło:
-
Bulletin of the Section of Logic; 2016, 45, 3/4
0138-0680
2449-836X - Pojawia się w:
- Bulletin of the Section of Logic
- Dostawca treści:
- Biblioteka Nauki
Artykuł