Abstract
We introduce the category of Heyting frames, those coherent frames L in which the compact elements form a Heyting subalgebra of L, and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological duality for distributive lattices and Heyting algebras. Math. Struct. Comput. Sci. 20(3), 359–393 (2010)
Bezhanishvili, G., Carai, L., Morandi, P.J.: Deriving Priestley duality and its generalizations from Pontryagin duality for semilattices (2022). Submitted. Available at arXiv:2207.13938
Bezhanishvili, G., Jansana, R.: Priestley style duality for distributive meet-semilattices. Studia Logica 98(1–2), 83–122 (2011)
Bezhanishvili, G., Jansana, R.: Esakia style duality for implicative semilattices. Appl. Categ. Struct. 21(2), 181–208 (2013)
Bezhanishvili, G., Moraschini, T., Raftery, J.G.: Epimorphisms in varieties of residuated structures. J. Algebra 492, 185–211 (2017)
Bhattacharjee, P.: Two spaces of minimal primes. J. Algebra Appl. 11(1), 1250014 (2012)
Birkhoff, G.: Lattice Theory. Vol. 25, 3rd edn. American Mathematical Society Colloquium Publications, Providence (1979)
Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides, vol. 35. The Clarendon Press, Oxford University Press, New York (1997)
Cornish, W.H.: On H. Priestley’s dual of the category of bounded distributive lattices. Mat. Vesnik 12(27)(4), 329–332 (1975)
Davey, B.A., Galati, J.C.: A coalgebraic view of Heyting duality. Studia Logica 75(3), 259–270 (2003)
Esakia, L.L.: Topological Kripke models. Soviet Math. Dokl. 15, 147–151 (1974)
Esakia, L.L.: Heyting Algebras: Duality Theory. Translated from the Russian by A. Evseev. Bezhanishvili, G., Holliday, W. (eds.) Trends in Logic, vol. 50. Springer, Berlin (2019)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York-Heidelberg (1977)
Iberkleid, W., McGovern, W.W.: A natural equivalence for the category of coherent frames. Algebra Universalis 62(2–3), 247–258 (2009)
Johnstone, P.T.: Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1982)
Köhler, P.: Varieties of Brouwerian algebras. Mitt. Math. Sem. Giessen (116), iii+83 (1975)
Köhler, P.: Brouwerian semilattices. Trans. Am. Math. Soc. 268(1), 103–126 (1981)
Köhler, P., Pigozzi, D.: Varieties with equationally definable principal congruences. Algebra Universalis 11(2), 213–219 (1980)
Nemitz, W.C.: Implicative semi-lattices. Trans. Am. Math. Soc. 117, 128–142 (1965)
Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel (2012)
Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)
Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 3(24), 507–530 (1972)
Priestley, H.A.: Ordered sets and duality for distributive lattices. In: Orders: Description and Roles (L’Arbresle, 1982), North-Holland Math. Stud., vol. 99, pp. 39–60. North-Holland, Amsterdam (1984)
Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics. Monografie Matematyczne, Tom 41. Państwowe Wydawnictwo Naukowe, Warsaw (1963)
Stone, M.H.: Topological representations of distributive lattices and Brouwerian logics. Čas. Mat. Fys. 67, 1–25 (1937)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Presented by N. Galatos.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
L. Carai acknowledges financial support from the Juan de la Cierva-Formación 2021 programme (FJC2021-046977-I) funded by MCIN/AEI/10.13039/501100011033 and by the European Union “NextGenerationEU”/PRTR.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) 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
Bezhanishvili, G., Carai, L. & Morandi, P.J. A frame-theoretic perspective on Esakia duality. Algebra Univers. 84, 30 (2023). https://doi.org/10.1007/s00012-023-00827-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-023-00827-3