Duality for bi-algebraic lattices belonging to the variety of $(0,1)$-lattices generated by the pentagon
Algebra i logika, Tome 63 (2024) no. 2, pp. 167-208
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According to G. Birkhoff, there is 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 is either a $2$-element chain or has uncountably many elements and is not distributive.
Keywords:
duality, bi-algebraic lattice, variety.
@article{AL_2024_63_2_a3,
author = {W. Dziobiak and M. V. Schwidefsky},
title = {Duality for bi-algebraic lattices belonging to the variety of $(0,1)$-lattices generated by the pentagon},
journal = {Algebra i logika},
pages = {167--208},
publisher = {mathdoc},
volume = {63},
number = {2},
year = {2024},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AL_2024_63_2_a3/}
}
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%0 Journal Article %A W. Dziobiak %A M. V. Schwidefsky %T Duality for bi-algebraic lattices belonging to the variety of $(0,1)$-lattices generated by the pentagon %J Algebra i logika %D 2024 %P 167-208 %V 63 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/AL_2024_63_2_a3/ %G ru %F AL_2024_63_2_a3
W. Dziobiak; M. V. Schwidefsky. Duality for bi-algebraic lattices belonging to the variety of $(0,1)$-lattices generated by the pentagon. Algebra i logika, Tome 63 (2024) no. 2, pp. 167-208. http://geodesic.mathdoc.fr/item/AL_2024_63_2_a3/