Abstract
In this paper we introduce Hilbert algebras with Hilbert–Galois connections (HilGC-algebras) and we study the Hilbert–Galois connections defined in Heyting algebras, called HGC-algebras. We assign a categorical duality between the category HilGC-algebras with Hilbert homomorphisms that commutes with Hilbert–Galois connections and Hilbert spaces with certain binary relations and whose morphisms are special functional relations. We also prove a categorical duality between the category of Heyting Galois algebras with Heyting homomorphisms that commutes with Hilbert–Galois connections and the category of spectral Heyting spaces endowed with a binary relation with certain special continuous maps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balbes, R., and P. H. Dwinger, Distributive Lattices, University of Missouri Press, 1974.
Bezhanishvili, G., N. Bezhanishvili, D. Gabelaia, and A. Kurz, Bitopological duality for distributive lattices and Heyting algebras, Mathematical Structures in Computer Science 20(3):359–393, 2010.
Birkhoff, G., Lattice Theory, 1st ed, American Mathematical Society Colloquium Publications AMS, Providence, RI 1940 (3rd ed, 1967).
Celani, S. A., A note on homomorphism of Hilbert algebras, International Journal of Mathematics and Mathematical Sciences 29(1):55–61, 2002.
Celani, S. A., Representation of Hilbert algebras and implicative semilattices, Central European Mathematical Journal 4:561–571, 2003.
Celani, S. A., L. M. Cabrer, and D. Montangie, Representation and duality for Hilbert algebras, Central European Journal of Mathematics 7(3):463–478, 2009.
Celani, S. A., and R. Jansana, A note on Hilbert algebras and their related generalized Esakia spaces, Order 33(3):429–458, 2016.
Celani, S. A., and D. Montangie, Hilbert algebras with supremum, Algebra Universalis 67(3):237–255, 2012.
Celani, S. A., and D. Montangie, Hilbert algebras with a necessity modal operator, Reports on Mathematical Logic 49:47–77, 2014.
Celani, S. A., and D. Montangie, Hilbert algebras with a modal operator \(\Diamond \), Studia Logica 103(3):639–662, 2015.
Chajda, I., R. Halas, and J. Kuhr, Semilattice structures, in Research and Exposition in Mathematics, vol. 30, Heldermann Verlag, 2007.
Dickmann, M., N. Schwartz, and M. Tressl, Spectral Spaces. New Monographs in Mathematics, vol. 35, Cambridge University Press, 2019.
Diego, A., Sur les algèbres de Hilbert, Hermann, Paris, Collection de Logique Math., Sèr. A 21, 1966.
Dzik, W., J. Jarvinen, and M. Kondo, Intuitionistic propositional logic with Galois connections, Logic Journal of the IGPL 18:837–858, 2010.
Dzik, W., J. Jarvinen, and M. Kondo, Characterizing intermediate tense logics in terms of Galois connections, Logic Journal of IGPL 22(6):992–1018, 2014.
Dzik, W., J. Jarvinen, and M. Kondo, Representing expansions of bounded distributive lattices with Galois connections in terms of rough sets, International Journal of Approximate Reasoning 55(1, Part 4):427–435, 2014.
Esakia, L., Topological Kripke models, Soviet Mathematics Doklady 15:147–151, 1974.
Esakia, L., Heyting Algebras I. Duality Theory (in Russian), Metsniereba, 1985.
Hochster, M., Prime ideal structure in commutative rings, Transactions of the American Mathematical Society 142:43–60, 1969.
Jarvinen, J., M. Kondo, and J. Kortelainen, Logics from Galois connections, International Journal of Approximate Reasoning 49:595–606, 2008.
Johnstone, P., Stone Spaces, Cambridge University Press, 1982.
Kowalski, T., Varieties of Tense Algebras, Reports on Mathematical Logic 32: 53–95, 1998.
Nemitz, W. C., Implicative semi-lattices, Transactions of the American Mathematical Society 117:128–142, 1965.
Ore, O., Galois connexions, Transactions of the American Mathematical Society 55:493–513, 1944.
Pacuit, E., Neighborhood Semantics for Modal Logic, Short Textbooks in Logic, Springer, 2017.
Priestley, H., Representation of distributive lattices by means of ordered Stone spaces, Bulletin of the London Mathematical Society 2:186–190, 1970.
Schmidt, J., Beitrage zur Filtertheorie, II, Mathematische Nachrichten 10:197–232, 1953.
Stone, M., Topological representation of distributive lattices and Brouwerian logics, Časopis pro pěstování matematiky a fysiky 67:1–25, 1937.
Thomason, S. K., Semantic analysis of tense logics, Journal of Symbolic Logic 37(1):150–158, 1972.
Acknowledgements
We would like to thank the referees for the comments and suggestions on the presentation of this paper. The authors acknowledge partial support of the project PICT 2019-00882 (2021-2023) CaToAM: triple abordaje semántico de las lógicas modales multivaluadas. The second author was supported by the MSCA-RISE - Marie Skłodowska-Curie Research and Innovation Staff Exchange (RISE) Project MOSAIC No. 101007627 (https://doi.org/10.3030/101007627) funded by Horizon 2020 of the European Union.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Yde Venema; Received October 15, 2020.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Celani, S.A., Montangie, D. Hilbert Algebras with Hilbert–Galois Connections. Stud Logica 111, 113–138 (2023). https://doi.org/10.1007/s11225-022-10019-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-022-10019-0