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On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

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Abstract

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.

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References

  1. Bellissima F.: Finitely generated free Heyting algebras. JSL 51(1), 152–165 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bezhanishvili G., Gehrke M., Mines R., Morandi P.J.: Profinite completions and canonical extensions of Heyting algebras. Order 23(2–3), 143–161 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blackburn P., de Rijke M., de Venema Y.: Modal Logic. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  4. Bezhanishvili, N.: Lattices of Intermediate and Cylindric Modal Logics. Doctoral thesis, Universiteit van Amsterdam (2006)

  5. Bezhanishvili G., Bezhanishvili N.: Profinite Heyting algebras. Order 25(3), 211–223 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Darnière, L.: Model-completion of Scaled Lattices. LAREMA-Preprint No. 191, Université d’Angers, mai (2004)

  7. Darnière, L., Junker, M.: Codimension and Pseudometric in (dual) Heyting Algebras. Algebra Universalis (to appear)

  8. de Jongh, D., Visser, A.: Embeddings of Heyting Algebras. In: Logic: from Foundations to Applications, pp. 187–213, Oxford University Press, New York (1996)

  9. Fitting M.: Intuitionistic Logic Model Theory and Forcing. North Holland, Amsterdam (1969)

    MATH  Google Scholar 

  10. Ghilardi S.: Free Heyting algebras as bi-Heyting algebras. C. R. Math. Rep. Acad. Sci. Can. 14(6), 240–244 (1992)

    MATH  MathSciNet  Google Scholar 

  11. Ghilardi, S.: Irreducible models and definable embeddings. In: Csirmaz, Gabbay, de Rijke (eds.) Logic Colloquium ’92, Studies in Logic, Language and Information, CSLI Publications, Stanford, pp. 95–113 (1995)

  12. Ghilardi S., Zawadowski M.: Model completions and r-Heyting categories. APAL 88, 27–46 (1997)

    MATH  MathSciNet  Google Scholar 

  13. Ghilardi, S., Zawadowski, M.: Sheaves, Games, and Model Completions, Trends in Logic col. 14, Kluwer Academic Publishers, Dordrecht 2002. APAL 88, 27–46 (1997)

  14. Grigolia R.: Free Algebras of Nonclassical Logics. Metsniereba Press, Tbilisiz (1987)

    Google Scholar 

  15. Grigolia R.: Free and projective Heyting and monadic Heyting algebras. In: Höhle, U., Klement, E.P. (eds) Non-Classical Logics and Their Applications to Fuzzy Subsets, pp. 33–52. Kluwer Academic Publisher, Dordrecht (1995)

    Google Scholar 

  16. Grigolia, R.: Free Heyting algebras and their automorphism groups. In: Proceedings of Institute of Cybernetics vol. 2(1–2) (2002)

  17. Hodges, W.: Model theory. In: Encyclopedia of Mathematics and its Applications vol. 42, Cambridge University Press, Cambridge (1993)

  18. Idziak P.: Elementary theory of free Heyting algebras. Rep. Math. Log. 23(1989), 71–73 (1990)

    Google Scholar 

  19. McKinsey J., Tarski A.: On closed elements in closure algebras. Ann. Math. 47(1), 122–162 (1946)

    Article  MathSciNet  Google Scholar 

  20. Pitts A.: On an interpretation of second order quantification in first order intuitionistic propositional logic. JSL 57(1), 33–52 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rybakov V.V.: The elementary theories of free topo-Boolean and pseudo-Boolean algebras. Mat. Zamet. 37(6), 797–802 (1985)

    MathSciNet  Google Scholar 

  22. Stone M.H.: Topological representations of distributive Lattices and Brouwerian Logics. Časopis Pro p̌estování Matematikyv a Fysiky 67, 1–25 (1937)

    Google Scholar 

  23. Urquhart A.: Free Heyting algebras. Algebra Univ. 3, 94–97 (1973)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences, Université d’Angers, 2 bd Lavoisier, 49045, Angers, France

    Luck Darnière

  2. Mathematisches Institut, Abteilung für Mathematische Logik, Albert–Ludwigs–Universität Freiburg, Eckerstraße 1, 79102, Freiburg, Germany

    Markus Junker

Authors
  1. Luck Darnière
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  2. Markus Junker
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Correspondence to Markus Junker.

Additional information

L. Darnière would like to thank the Universität Freiburg for inviting him in July 2008.

M. Junker would like to thank the Université d’Angers for supporting him as an invited professor in march 2005, when main parts of this work were done.

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Darnière, L., Junker, M. On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond. Arch. Math. Logic 49, 743–771 (2010). https://doi.org/10.1007/s00153-010-0194-7

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  • DOI: https://doi.org/10.1007/s00153-010-0194-7

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