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Profinite Completions and Canonical Extensions of Heyting Algebras

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Abstract

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion \(\widehat{H}\) of a Heyting algebra H, and characterize the dual space of \(\widehat{H}\). We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical.

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References

  1. Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia, Missouri (1974)

    MATH  Google Scholar 

  2. Banaschewski, B.: Coherent frames. Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices. Lecture Notes in Math., vol. 871, pp. 1–11. Springer, Berlin Heidelberg New York (1981)

    Google Scholar 

  3. Bezhanishvili, G.: Varieties of monadic Heyting algebras. II. Duality Theory. Stud. Log. 62(1), 21–48 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bezhanishvili, G., Grigolia, R.: Locally finite varieties of Heyting algebras. Algebra Univers. 54(4), 465–473 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blok, W.: Varieties of interior algebras. PhD thesis, University of Amsterdam, The Netherlands (1976)

  6. Chagrov, A., Zakharyaschev, M.: Modal Logic Oxford Logic Guides, vol. 35. Clarendon, Oxford UK (1997)

    Google Scholar 

  7. Choe, T.: Partially ordered topological spaces. An. Acad. Bras. Ciênc. 51(1), 53–63 (1979)

    MATH  MathSciNet  Google Scholar 

  8. Cornish, W.: On H. Priestley’s dual of the category of bounded distributive lattices. Mat. Vesn. 12(27) (no. 4), 329–332 (1975)

    MATH  MathSciNet  Google Scholar 

  9. Davey, B., Galati, J.: A coalgebraic view of Heyting duality. Stud. Log. 75(3), 259–270 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Engelking, R.: General Topology. PWN–Polish Scientific Publisher, Warsaw, Poland (1977)

    Google Scholar 

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

    MATH  Google Scholar 

  12. Esakia, L.: Heyting Algebras I. Duality Theory (Russian). Metsniereba, Tbilisi, Georgia (1985)

    Google Scholar 

  13. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 40(2), 207–215 (1994)

    MATH  Google Scholar 

  15. Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Jpn. 52(2), 197–213 (2000)

    MATH  Google Scholar 

  16. Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94(1), 13–45 (2004)

    MATH  MathSciNet  Google Scholar 

  17. Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, New York; Princeton University Press, New Jersey (1960)

  18. Grätzer, G.: Universal Algebra. Springer, Berlin Heidelberg New York (1979)

    MATH  Google Scholar 

  19. Hansoul, G.: The Stone–Čech compactification of a pospace. Lectures in Universal Algebra, Szeged, 1983, Colloq. Math. Soc. János Bolyai, vol. 43, pp. 161–176. North-Holland, Amsterdam (1986)

    Google Scholar 

  20. Harding, J.: On profinite completions and canonical extensions. Algebra Universalis (to appear)

  21. Jónsson, B.: On the canonicity of Sahlqvist identities. Stud. Log. 53(4), 473–491 (1994)

    Article  MATH  Google Scholar 

  22. Jónsson, B., Tarski, A.: Boolean algebras with operators. I. Am. J. Math. 73, 891–939 (1951)

    Article  MATH  Google Scholar 

  23. Maksimova, L.: Pretabular superintuitionistic logics. Algebra and Logic 11, 308–314 (1972)

    Article  MathSciNet  Google Scholar 

  24. Nachbin, L.: Topology and Order. Van Nostrand, New York; Princeton University Press, New Jersey (1965)

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

    MATH  MathSciNet  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, Mai Gehrke, Ray Mines & Patrick J. Morandi

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  1. Guram Bezhanishvili
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  2. Mai Gehrke
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  3. Ray Mines
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  4. Patrick J. Morandi
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Correspondence to Patrick J. Morandi.

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Bezhanishvili, G., Gehrke, M., Mines, R. et al. Profinite Completions and Canonical Extensions of Heyting Algebras. Order 23, 143–161 (2006). https://doi.org/10.1007/s11083-006-9037-x

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