Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
MetadataPokaż pełny rekord
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. Jakubík from . Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S*7 ≪ 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.
Wyświetlanie pozycji powiązanych tytułem, autorstwem i tematem.
Decomposition of Congruence Modular Algebras into Atomic, Atomless Locally Uniform and Anti-Uniform Parts Staruch, Bogdan; Staruch, Bożena (Wydawnictwo Uniwersytetu Łódzkiego, 2016)We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition (called a star-decomposition) is based on the properties of the congruence lattices of algebras. We ...
Assaye, Berhanu; Alemneh, Mihret; Tefera, Gerima (Wydawnictwo Uniwersytetu Łódzkiego, 2018)The paper introduces the concept of B-Almost distributive fuzzy lattice (BADFL) in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also ...
Staruch, Bogdan (Wydawnictwo Uniwersytetu Łódzkiego, 2016)We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of ...