Natural Duality as a Tool to Study Algebras Arising from Logics

  • Philippe Niederkorn
Conference paper
Part of the Advances in Soft Computing book series (AINSC, volume 11)


MV-algebras are the algebraic counterpart of Lukasiewicz’s infinite-valued logic, just as Boolean algebras correspond to the classical propositional calculus. The finitely generated subvarieties of the variety M of all MV-algebras are generated by a finite number of finite chains.

We present Davey and Werner’s theory of natural duality, illustrated by its application to a few classes of algebras arising from classical and non-classical logics. We insist on the subvarieties of M generated by one finite chain, for which some simple applications of the dualities are proposed.


Boolean Algebra Free Algebra Natural Duality Strong Duality Unary Relation 


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  1. 1.
    R. Balbes and P. Dwinger, “Distributive lattices”, University of Missouri Press, 1974.Google Scholar
  2. 2.
    C. C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D. M. Clark and B. A. Davey, “Natural Dualities for the Working Algebraist”, Cambridge University Press, 1998.Google Scholar
  4. 4.
    B. A. Davey and H. A. Priestley, “Introduction to Lattices and Order”, Cambridge University Press, 1990.Google Scholar
  5. 5.
    B. A. Davey and H. Werner, Dualities and equivalences for varieties of algebras, in “Colloquia mathematica societatis Janos Bolyai”, Vol. 33, North-Holland, 1983.Google Scholar
  6. 6.
    A. Di Nola, R. Grigolia and G. Panti, Finitely generated free MV-algebras and their automorphism groups, Studia Logica 61 (1998), 65–78.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    A. Di Nola and R. Grigolia, Projective MV-algebras and their automorphism groups, to appearGoogle Scholar
  8. 8.
    A. M. Gleason, Projective topological spaces, Ill. J. Math. 2 (1958), 482–489.MathSciNetMATHGoogle Scholar
  9. 9.
    R. Grigolia, “Free algebras of non-classical logics”, Metsniereba, Tbilisi, 1987.Google Scholar
  10. 10.
    T. K. Hu, On the topological duality for primal algebra theory, Alg. Univ. 1 (1971), 152–154.MATHCrossRefGoogle Scholar
  11. 11.
    P. Niederkorn, Natural dualities for varieties of MV-algebras. I, to appear in J. Math. Anal. Appl. Google Scholar
  12. 12.
    H. A. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186–190.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    H. A. Priestley, Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 24 (1972), 507–530.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    H. Rasiowa, “An Algebraic Approach to Non-Classical Logics”, Studies in logic and the foundations of mathematics (volume 78), North-Holland/American Elsevier, 1974.Google Scholar
  15. 15.
    M. H. Stone, The Theory of representations for Boolean Algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Philippe Niederkorn
    • 1
  1. 1.Algebra and LogicUniversity of LiègeBelgium

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