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Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon. / Dziobiak, W.; Schwidefsky, M. V.

In: Algebra and Logic, Vol. 63, No. 2, 31.01.2024, p. 114-140.

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Dziobiak W, Schwidefsky MV. Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon. Algebra and Logic. 2024 Jan 31;63(2):114-140. doi: 10.1007/s10469-025-09776-3

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@article{117883f0e50b4f62b3cd6617e3ef3b1d,
title = "Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon",
abstract = "According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.",
keywords = "bi-algebraic lattice, duality, variety",
author = "W. Dziobiak and Schwidefsky, {M. V.}",
note = "The study was carried out as part of the state assignment to Sobolev Institute of Mathematics SB RAS (project FWNF-2022-0012) and supported by Russian Science Foundation (project No. 22-21-00104). Dziobiak, W. Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon / W. Dziobiak, M. V. Schwidefsky // Algebra and Logic. – 2024. – Vol. 63, No. 2. – P. 114-140. – DOI 10.1007/s10469-025-09776-3.",
year = "2024",
month = jan,
day = "31",
doi = "10.1007/s10469-025-09776-3",
language = "English",
volume = "63",
pages = "114--140",
journal = "Algebra and Logic",
issn = "0002-5232",
publisher = "Springer US",
number = "2",

}

RIS

TY - JOUR

T1 - Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon

AU - Dziobiak, W.

AU - Schwidefsky, M. V.

N1 - The study was carried out as part of the state assignment to Sobolev Institute of Mathematics SB RAS (project FWNF-2022-0012) and supported by Russian Science Foundation (project No. 22-21-00104). Dziobiak, W. Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon / W. Dziobiak, M. V. Schwidefsky // Algebra and Logic. – 2024. – Vol. 63, No. 2. – P. 114-140. – DOI 10.1007/s10469-025-09776-3.

PY - 2024/1/31

Y1 - 2024/1/31

N2 - According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.

AB - According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.

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KW - duality

KW - variety

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