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An algebraic approach to subframe logics. Intuitionistic case. (English) Zbl 1123.03055

A nucleus on a meet semilattice \(M\) with \(\top\) is a unary operator \(j : M \to M\) satisfying the conditions \(a \leq j(a)\), \(j(j(a)) \leq j(a)\) and \(j(a \wedge b) = j(a) \wedge j(b)\). If \(M\) has the least element \(\bot\), a nucleus \(j\) is called dense if \(j(\bot) = \bot\). In this paper, the authors develop a duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces, and they prove that these binary relations are in 1-1 correspondence with subframes of Heyting spaces. They introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear if and only if it is a subframe variety, and that it is dense nuclear if and only if it is a cofinal subframe variety. Finally, they prove that every nuclear and dense nuclear variety of Heyting algebras is generated by its finite members.

MSC:

03G25 Other algebras related to logic
06D20 Heyting algebras (lattice-theoretic aspects)
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[1] Balbes, Raymond; Dwinger, Philip, Distributive Lattices (1974), University of Missouri Press: University of Missouri Press Columbia, MO · Zbl 0321.06012
[2] Barr, Michael, On categories with effective unions, (Categorical Algebra and its Applications. Categorical Algebra and its Applications, Louvain-La-Neuve, 1987. Categorical Algebra and its Applications. Categorical Algebra and its Applications, Louvain-La-Neuve, 1987, Lecture Notes in Math., vol. 1348 (1988), Springer: Springer Berlin), 19-35 · Zbl 0661.18002
[3] Beazer, R.; Macnab, D. S., Modal extensions of Heyting algebras, Colloq. Math., 41, 1, 1-12 (1979) · Zbl 0436.06010
[4] Bozzi, Silvio; Meloni, Gian Carlo, Completezza proposizionale per l’operatore modale “accade localmente che”, Boll. Unione Mat. Ital. A (5), 14, 3, 489-497 (1977), (in Italian) · Zbl 0376.02021
[5] Bozzi, Silvio; Meloni, Gian Carlo, Representation of Heyting algebras with covering and propositional intuitionistic logic with local operator, Boll. Unione Mat. Ital. A (5), 17, 3, 436-442 (1980) · Zbl 0454.03008
[6] Cassidy, C.; Hébert, M.; Kelly, G. M., Reflective subcategories, localizations and factorization systems, J. Aust. Math. Soc. Ser. A, 38, 3, 287-329 (1985) · Zbl 0573.18002
[7] Chagrov, Alexander; Zakharyaschev, Michael, (Modal Logic. Modal Logic, Oxford Logic Guides, vol. 35 (1997), Clarendon Press: Clarendon Press Oxford) · Zbl 0871.03007
[8] Diego, Antonio, Sur les algèbres de Hilbert (Translated from the Spanish by Luisa Iturrioz), (Collection de Logique Mathématique, Sér. A, Fasc. XXI (1966), Gauthier-Villars: Gauthier-Villars Paris)
[9] Esakia, L. L., Topological Kripke models, Soviet Math. Dokl., 15, 147-151 (1974) · Zbl 0296.02030
[10] Esakia, L. L., Heyting algebras. I. Duality theory (1985), Metsniereba: Metsniereba Tbilisi, (in Russian) · Zbl 0601.06009
[11] Fine, Kit, Logics containing K4. II, J. Symbolic Logic, 50, 3, 619-651 (1985) · Zbl 0574.03008
[12] Ghilardi, Silvio; Meloni, Gian Carlo, Models with coverings for intuitionistic and local propositional logic, (Proceedings of the Conference on Mathematical Logic, Vol. 2. Proceedings of the Conference on Mathematical Logic, Vol. 2, Siena, 1983-1984 (1985), Univ. Siena: Univ. Siena Siena), 333-338 · Zbl 0598.03047
[13] Goldblatt, Robert I., Grothendieck topology as geometric modality, Z. Math. Logik Grundlag. Math., 27, 6, 495-529 (1981) · Zbl 0474.03018
[14] Jankov, V. A., On the relation between deducibility in intuitionistic propositional calculus and finite implicative structures, Dokl. Akad. Nauk SSSR, 151, 1293-1294 (1963), (in Russian)
[15] Joyal, André; Tierney, Myles, An extension of the Galois theory of Grothendieck, (Mem. Amer. Math. Soc., vol. 51 (1984)) · Zbl 0541.18002
[16] Johnstone, Peter T., Stone spaces, (Cambridge Studies in Advanced Mathematics, vol. 3 (1982), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0499.54001
[17] Johnstone, Peter T., (Sketches of an elephant: A topos theory compendium, Vol. 1. Sketches of an elephant: A topos theory compendium, Vol. 1, Oxford Logic Guides, vol. 43 (2002), The Clarendon Press Oxford University Press: The Clarendon Press Oxford University Press New York) · Zbl 1071.18001
[18] Köhler, Peter, Brouwerian semilattices, Trans. Amer. Math. Soc., 268, 1, 103-126 (1981) · Zbl 0473.06003
[19] Macnab, D. S., Modal operators on Heyting algebras, Algebra Universalis, 12, 1, 5-29 (1981) · Zbl 0459.06005
[20] McKay, C. G., The decidability of certain intermediate propositional logics, J. Symbolic Logic, 33, 258-264 (1968) · Zbl 0175.27103
[21] Nemitz, William C., Implicative semi-lattices, Trans. Amer. Math. Soc., 117, 128-142 (1965) · Zbl 0128.24804
[22] Rasiowa, H.; Sikorski, R., The mathematics of metamathematics, (Monografie Matematyczne, Tom 41 (1963), Państwowe Wydawnictwo Naukowe: Państwowe Wydawnictwo Naukowe Warsaw) · Zbl 0122.24311
[23] Wolter, Frank, The structure of lattices of subframe logics, Ann. Pure Appl. Logic, 86, 1, 47-100 (1997) · Zbl 0878.03015
[24] Wroński, Andrzej, Intermediate logics and the disjunction property, Rep. Math. Logic, 1, 39-51 (1973) · Zbl 0308.02027
[25] Zakharyaschev, Michael V., Syntax and semantics of superintuitionistic logics, Algebra Logic, 28, 262-282 (1989) · Zbl 0708.03011
[26] Zakharyaschev, Michael V., Canonical formulas for K4. II. Cofinal subframe logics, J. Symbolic Logic, 61, 2, 421-449 (1996) · Zbl 0884.03014
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