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Esakia Style Duality for Implicative Semilattices

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Abstract

We develop a new duality for implicative semilattices, generalizing Esakia duality for Heyting algebras. Our duality is a restricted version of generalized Priestley duality for distributive semilattices, and provides an improvement of Vrancken-Mawet and Celani dualities. We also show that Heyting algebra homomorphisms can be characterized by means of special partial functions between Esakia spaces. On the one hand, this yields a new duality for Heyting algebras, which is an alternative to Esakia duality. On the other hand, it provides a natural generalization of Köhler’s partial functions between finite posets to the infinite case.

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References

  1. Banaschewski, B., Bruns, G.: The fundamental duality of partially ordered sets. Order 5(1), 61–74 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bezhanishvili, G., Bezhanishvili, N.: An algebraic approach to canonical formulas: intuitionistic case. Rev. Symb. Log. 2(3), 517–549 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bezhanishvili, G., Jansana, R.: Duality for distributive and implicative semi-lattices. Preprints of University of Barcelona Research Group in Non-Classical Logics. http://www.mat.ub.edu/~logica/docs/BeJa08-m.pdf (2008)

  5. Bezhanishvili, G., Jansana, R.: Priestley style duality for distributive meet-semilattices. Stud. Log. 98(1–2), 83–122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Celani, S.A.: Topological representation of distributive semilattices. Sci. Math. Jpn. 58(1), 55–65 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Celani, S.A.: Representation of Hilbert algebras and implicative semilattices. Cent. Eur. J. Math. 1(4), 561–572 (electronic, 2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cornish, W.H.: On H. Priestley’s dual of the category of bounded distributive lattices. Mat. Vesnik 12(27)(4), 329–332 (1975)

    Google Scholar 

  9. Erné, M.: Algebraic ordered sets and their generalizations. Algebras and Orders (Montreal, PQ, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 389, pp. 113–192. Kluwer Acad. Publ., Dordrecht (1993)

  10. Erné, M.: Choiceless, pointless, but not useless: dualities for preframes. Appl. Categ. Struct. 15(5–6), 541–572 (2007)

    Article  MATH  Google Scholar 

  11. Esakia, L.L.: Topological Kripke models. Sov. Math., Dokl. 15, 147–151 (1974)

    MATH  Google Scholar 

  12. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: A Compendium of Continuous Lattices. Springer-Verlag, Berlin (1980)

    Book  MATH  Google Scholar 

  13. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous lattices and domains. Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  14. Grätzer, G.: General Lattice Theory (2nd ed.). Birkhäuser Verlag, Basel (1998, New appendices by the author with Davey, B.A., Freese, R., Ganter, B., Greferath, M., Jipsen, P., Priestley, H.A., Rose, H., Schmidt, E.T., Schmidt, S.E., Wehrung, F., Wille, R.)

  15. Hansoul, G., Poussart, C.: Priestley duality for distributive semilattices. Bull. Soc. Roy. Sci. Liège 77, 104–119 (2008)

    MathSciNet  Google Scholar 

  16. Hofmann, K.H., Watkins, F.: The spectrum as a functor. In: Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices, Lecture Notes in Math., vol. 871, pp. 249–263. Springer-Verlag, Berlin (1981)

    Google Scholar 

  17. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  18. Köhler, P.: Brouwerian semilattices. Trans. Am. Math. Soc. 268(1), 103–126 (1981)

    Article  MATH  Google Scholar 

  19. Köhler, P., Pigozzi, D.: Varieties with equationally definable principal congruences. Algebra Univers. 11(2), 213–219 (1980)

    Article  MATH  Google Scholar 

  20. Nemitz, W.C.: Implicative semi-lattices. Trans. Am. Math. Soc. 117, 128–142 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  21. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  22. Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 24(3), 507–530 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40(1), 37–111 (1936)

    Google Scholar 

  24. Stone, M.H.: Topological representation of distributive lattices and Brouwerian logics. Časopis Pešt. Mat. Fys. 67, 1–25 (1937)

    Google Scholar 

  25. Vrancken-Mawet, L.: Dualité pour les demi-lattis de Brouwer. Bull. Soc. Roy. Sci. Liège 55(2), 346–352 (1986)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003-8001, USA

    Guram Bezhanishvili

  2. Dept. Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001, Barcelona, Spain

    Ramon Jansana

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  1. Guram Bezhanishvili
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  2. Ramon Jansana
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Correspondence to Guram Bezhanishvili.

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Bezhanishvili, G., Jansana, R. Esakia Style Duality for Implicative Semilattices. Appl Categor Struct 21, 181–208 (2013). https://doi.org/10.1007/s10485-011-9265-0

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  • DOI: https://doi.org/10.1007/s10485-011-9265-0

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