Abstract
Generalized Esakia spaces are the topological duals of bounded implicative semilattices in the duality studied by G. Bezhanishvili and R. Jansana. We study the relation between a Hilbert algebra and the generalized Esakia space dual to its free implicative semilattice extension. To establish the relation we introduce a category whose objects are a generalized Esakia space together with a family of clopen up-sets that constitutes a subalgebra of the implication fragment of the Heyting algebra of the up-sets of the generalized Esakia space.
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References
Bezhanishvili, G. (ed.): Leo Esakia on duality in modal and intuitionistic logics. Springer (2014)
Bezhanishvili, G., Jansana, R.: Duality for distributive and implicative semi-lattices, http://www.mat.ub.edu/logica/ppts.html
Bezhanishvili, G., Jansana, R.: Priestley duality for distributive meet-semilattices, Studia Logica, special issue “Algebras Related to Non-classical Logics”, 98, pp. 83–123 (2011)
Bezhanishvili, G., Jansana, R.: Esakia style duality for implicative semilattices. Appl. Categ. Struct. 21–2, 181–208 (2013)
Celani, S.: A note on homomorphisms of Hilbert algebras. Int. J. Math. Math. Sci. 29, 55–61 (2002)
Celani, S.: Representation of Hilbert algebras and implicative semilattices. Cent. Eur. J. Math., 561–572 (2003)
Celani, S., Cabrer, L., Montangie, D.: Representation and duality for Hilbert algebras. Cent. Eur. J. Math., 463–478 (2009)
Celani, S., Jansana, R.: On the free implicative semilattice extension of a Hilbert algebra. Math. Log. Q. 58, 188–207 (2012)
David, E., Erné, M.: Ideal completion and Stone representation of ideal-distributive ordered sets. Topol. Appl. 44, 95–113 (1992)
Diego, A.: Sur les algébres de Hilbert, Collection de Logique Mathématique, Sér. A, fasc, vol. 21. Gauthier-Villars, Paris (1966)
Erné, M.: Prime and maximal ideals of partially ordered sets. Math. Slovaca 56, 1–22 (2006)
Frink, O.: Ideals in partially ordered sets. Amer. Math. Monthly 61, 223–234 (1954)
Köhler, P.: Brouwerian semilattices. Trans. Am. Math. Soc. 268, 103–126 (1981)
Monteiro, A.: Sur les algébres de Heyting symétriques. Portugal Math. 39, 1–237 (1980)
Nemitz, W.C.: Implicative semi-lattices. Trans. Am. Math. Soc. 117, 128–142 (1965)
Porta, H.: Sur quelques algèbres de la logique. Portugal. Math 40, 41–77 (1981)
Davey, B.A., Priestley, H.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Rasiowa, H.: An algebraic approach to non-classical logics. North-Holland, Amsterdam (1974)
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Celani, S.A., Jansana, R. A Note on Hilbert Algebras and Their Related Generalized Esakia Spaces. Order 33, 429–458 (2016). https://doi.org/10.1007/s11083-015-9378-4
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DOI: https://doi.org/10.1007/s11083-015-9378-4