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.

KW - bi-algebraic lattice

KW - duality

KW - variety

UR - https://www.mendeley.com/catalogue/d1113d13-b6bb-3cf2-9aba-bf53b289a656/

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-105008505400&origin=inward&txGid=6ca0a834c2bca047b1be29aff1c5a63a

UR - https://www.elibrary.ru/item.asp?id=81349191

U2 - 10.1007/s10469-025-09776-3

DO - 10.1007/s10469-025-09776-3

M3 - Article

VL - 63

SP - 114

EP - 140

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 2

ER -