Categorical Dualities for Some Two Categories of Lattices: An Extended Abstract

Wiesław Dziobiak; Marina Schwidefsky

doi:10.18778/0138-0680.2022.14

Authors

Wiesław Dziobiak University of Puerto Rico, Mayagüez Campus, 00681-9018, Mayagüez, Puerto Rico, USA Author
Marina Schwidefsky Sobolev Institute of Mathematics SB RAS, Laboratory of Logical Structures, 630090, Acad. Koptyug prosp. 4, Novosibirsk, Russia Author https://orcid.org/0000-0003-4804-8073

DOI:

https://doi.org/10.18778/0138-0680.2022.14

Keywords:

categorical duality, bi-algebraic lattice, bounded lattice, quasivariety lattice

Abstract

The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the presented dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive.

References

M. E. Adams, K. V. Adaricheva, W. Dziobiak, A. V. Kravchenko, Open questions related to the problem of Birkhoff and Maltsev, Studia Logica, vol. 78 (2004), pp. 357–378, DOI: https://doi.org/10.1007/s11225-005-7378-x DOI: https://doi.org/10.1007/s11225-005-7378-x

M. E. Adams, W. Dziobiak, Finite-to-finite universal quasivarieties are Q- universal, Algebra Universalis, vol. 46 (2001), pp. 253–283, DOI: https://doi.org/10.1007/PL00000343 DOI: https://doi.org/10.1007/PL00000343

M. E. Adams, W. Dziobiak, A. V. Kravchenko, M. V. Schwidefsky, Complete homomorphic images of the quasivariety lattices of locally finite quasivarieties (2020).

M. E. Adams, V. Koubek, J. Sichler, Homomorphisms and endomorphisms of distributive lattices, Houston Journal of Mathematics, vol. 11 (1984), pp. 129–145, DOI: https://doi.org/10.2307/1999472 DOI: https://doi.org/10.2307/1999472

W. H. Cornish, On H. Priestley’s dual of the category of bounded distributive lattices, Matematiˇcki Vesnik, vol. 12 (1975), pp. 329–332.

Y. L. Ershov, Solimit points and u-extensions, Algebra and Logic, vol. 56 (2017), pp. 295–301, DOI: https://doi.org/10.1007/s10469-017-9450-9 DOI: https://doi.org/10.1007/s10469-017-9450-9

Y. L. Ershov, M. V. Schwidefsky, To the spectral theory of partially ordered sets, Siberian Mathematical Journal, vol. 60 (2019), pp. 450–463, DOI: https://doi.org/10.1134/S003744661903008X DOI: https://doi.org/10.1134/S003744661903008X

Y. L. Ershov, M. V. Schwidefsky, To the spectral theory of partially ordered sets. II, Siberian Mathematical Journal, vol. 61 (2020), pp. 453—462, DOI: https://doi.org/10.1134/S0037446620030064 DOI: https://doi.org/10.1134/S0037446620030064

R. Freese, J. B. Nation, J. Ježek, Free Lattices, no. 42 in Mathematical Surveys and Monographs, American Mathematical Society, Providence (1995). DOI: https://doi.org/10.1090/surv/042

P. Goralčík, V. Koubek, J. Sichler, Universal varieties of (0,1)-lattices, Canadian Journal of Mathematics, vol. 42 (1990), pp. 470–490, DOI: https://doi.org/10.4153/CJM-1990-024-0 DOI: https://doi.org/10.4153/CJM-1990-024-0

V. A. Gorbunov, Algebraic Theory of Quasivarieties, Siberian School of Algebra and Logic, Plenum, Consultants Bureau, New York (1998).

A. P. Huhn, Schwach distributive Verbände. I, Acta Scientiarum Mathematicarum (Szeged), vol. 33 (1972), pp. 297–305.

M. A. Moshier, P. Jipsen, Topological duality and lattice expansions, I: A topological construction for canonical extensions, Algebra Universalis, vol. 71 (2014), pp. 109–126, DOI: https://doi.org/10.1007/s00012-014-0275-2 DOI: https://doi.org/10.1007/s00012-014-0267-2

J. B. Nation, An approach to lattice varieties of finite height, Algebra Universalis, vol. 27 (1990), pp. 521–543, DOI: https://doi.org/10.1007/BF01188998 DOI: https://doi.org/10.1007/BF01188998

H. A. Priestley, Ordered topological spaces and representation of distributive lattices, Proceedings of the London Mathematical Society, vol. 24 (1972), pp. 507–530, DOI: https://doi.org/10.1112/plms/s3-24.3.507 DOI: https://doi.org/10.1112/plms/s3-24.3.507

M. H. Stone, Topological representation of distributive lattices and Brouwerian logics, Časopis pro pěstování matematiky a fysiky, vol. 67 (1938), pp. 1–25. DOI: https://doi.org/10.21136/CPMF.1938.124080