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Hilbert Algebras with a Modal Operator \({\Diamond}\)

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Abstract

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie (2012). In this paper we shall introduce and study the variety of \({H_{\Diamond}^{\vee}}\)-algebras, which are Hilbert algebras with supremum endowed with a modal operator \({\Diamond}\). We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in Celani and Montangie (2012). We will consider some particular varieties of \({H_{\Diamond}^{\vee}}\)-algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic \({\mathbf{IntK}_{\Diamond}}\). We also determine the congruences of \({H_{\Diamond}^{\vee}}\)-algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible \({H_{\Diamond}^{\vee }}\)-algebras.

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References

  1. Bezhanisvili G.: Varieties of monadic Heyting algebras I. Studia Logica 61, 367–402 (1998)

    Article  Google Scholar 

  2. Bezhanishvili G., Jansana R.: Priestley style duality for distributive meet-semilattices. Studia Logica 98, 83–123 (2011)

    Article  Google Scholar 

  3. Bezhanishvili G., Jansana R.: Esakia style duality for implicative semi-lattices. Applied Categorical Structures 21(2), 181–208 (2013)

    Article  Google Scholar 

  4. Bozic M., Dosen K.: Models for normal intuitionistic modal logics. Studia Logica 44, 217–245 (1984)

    Article  Google Scholar 

  5. Celani S.A.: A note on homomorphism of Hilbert algebras. International Journal of Mathematics and Mathematical Sciences 29(1), 55–61 (2002)

    Article  Google Scholar 

  6. Celani S.A.: Representation of Hilbert algebras and implicative semilattices. Central European Mathematical Journal 4, 561–571 (2003)

    Article  Google Scholar 

  7. Celani S. A., Topological representation of distributive semilattices. Scientiae Mathematicae Scientiae Mathematicae Japonicae, Vol. 58, No. 1., 55-65, 2003.

  8. Celani, S. A., and L. M. Cabrer, Topological duality for Tarski algebras, Algebra Universalis 58(1): 73–94, 2008.

  9. Celani, S. A., L. M. Cabrer, and D. Montangie, Topological duality for Hilbert algebras, Central European Journal of Mathematics 7(3): 463–478, 2009.

  10. Celani, S. A., and D. Montangie, Hilbert algebras with supremum, Algebra Universalis 67(3): 237–255, 2012.

  11. Celani S.A., Montangie D.: Hilbert algebras with a necessity modal operator. Reports on Mathematical Logic 49, 47–77 (2014)

    Google Scholar 

  12. Celani S.A., Jansana R.: On the free implicative semilattice extension of a Hilbert algebra. Mathematical Logic Quarterly 58(3), 1–20 (2012)

    Article  Google Scholar 

  13. Celani, S. A., and R. Jansana, A topological duality for Hilbert algebras, Manuscript, 2012.

  14. Celani S.A.: Notes in bounded Hilbert algebras with supremum. Acta Scientiarum Mathematicarum (Szeged) 80, 3–19 (2014)

    Article  Google Scholar 

  15. Celani S.A., Calomino I.: Some remarks on distributive semilattices. Commentationes Mathematicae Universitatis Carolinae 54(3), 407–428 (2013)

    Google Scholar 

  16. Chajda, I., R. Halaš, and J. Kühr, Semilattice structures. Research and Exposition in Mathematics, Vol. 30, Heldermann Verlag, Lemgo, 2007.

  17. Cignoli R.: Quantifiers on distributive lattices. Discrete Mathematics 96, 183–197 (1991)

    Article  Google Scholar 

  18. Diego, A., Sur les algèbres de Hilbert, Colléction de Logique Mathèmatique, serie A, Vol. 21, Gouthier-Villars, Paris, 1966.

  19. Dosen K.: Models for stronger normal intuitionistic modal logics. Studia Logica 44(1), 39–70 (1983)

    Article  Google Scholar 

  20. Fischer Servi, G., On modal logics with an intuitionistic base, Studia Logica 36(2): 141–149, 1977.

  21. Fischer Servi, G., Axiomatizations for some intuitionistic modal logics, Rendiconti del Seminario Matematico dell’ Università à Politecnica di Torino 42(3): 179–194, 1984.

  22. Grätzer, G., General Lattice Theory, Birkhäuser Verlag, Basel, 1998.

  23. Gluschankof D., Tilli M.: Maximal deductive systems and injective objects in the category of Hilbert algebras. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 34(3), 213–220 (1988)

    Article  Google Scholar 

  24. Hansoul, G., and C. Poussart, Priestley duality for distributive semilattices, Bulletin de la Société Royale des Sciences de Liège 77: 104–119, 2008.

  25. Monteiro, A., Sur les algèbres de Heyting symétriques, Portugaliae Mathematica 39(fasc.): 1–4, 1980.

  26. Palmigiano, A., Dualities for Some Intuitionistic Modal Logics, Dick de Jongh’s Liber Amicorum, ILLC Publications, Amsterdam, 2004.

  27. Priestley H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society 2, 186–190 (1970)

    Article  Google Scholar 

  28. Orlowska E., Rewitzky I.: Discrete dualities for heyting algebras with operators. Fundamenta Infromaticae 81, 275–295 (2007)

    Google Scholar 

  29. Rueda, L., Linear Heyting algebras with a quantifier, Annals of Pure and Applied Logic 108(1–3) (3): 327–343, 2001.

  30. Stone, M., Topological representation of distributive lattices and Brouwerian logics, Časopis pro Peštováni matematiky a Fysiky, roč 67: 1–25, 1937.

  31. Sotirov, V. H., Modal theories with intuitionistic logic, In Mathematical Logic, Proceedings of the Conference on Mathematical Logic, Dedicated to the Memory of A. A. Markov (1903–1979), Sofia, 22–23 September 1980, 1984, pp. 139–171.

  32. Wolter, F., and M. Zakharyaschev, Intuitionistic modal logic, In Logic and foundations of mathematics (Florence, 1995), Vol. 280 of Synthese Library, Kluwer, Dordrecht, 1999, pp. 227–238.

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Authors and Affiliations

  1. CONICET and Departamento de Matemática, Facultad de Ciencias Exactas Universidad Nacional del Centro, Pinto 399, 7000, Tandil, Argentina

    Sergio A. Celani

  2. Departamento de Matemáticas, Facultad de Economia y Administracion, Universidad Nacional del Comahue, Buenos Aires 1400, 8300, Neuquén, Argentina

    Daniela Montangie

Authors
  1. Sergio A. Celani
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  2. Daniela Montangie
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Correspondence to Sergio A. Celani.

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Celani, S.A., Montangie, D. Hilbert Algebras with a Modal Operator \({\Diamond}\) . Stud Logica 103, 639–662 (2015). https://doi.org/10.1007/s11225-014-9583-y

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