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Some Invariant Skeletons for -u Groups and MV-Algebras

  • Antonio Di Nola
  • Giacomo Lenzi
  • Anna Carla Russo
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Abstract

In this paper we study some invariants for MV-algebras and thanks to Mundici’s equivalence we transfer these invariants to -groups with strong unit. In particular, we prove that, as it happens to MV-algebras, every -u group has two families of skeletons, which we call the n-skeletons and the \({}_{n}^{\omega }\)-skeletons. Then we study the classes of -u groups (and of MV-algebras) which coincide with the union of such skeletons, called here ω-skeletal and \({}_{\omega }^{\omega }\)-skeletal -u groups (resp. MV-algebras). We also analyze the problem of axiomatizing in terms of geometric theories or theories of presheaf type these classes of -u groups (and of MV-algebras).

Keywords

MV-algebra Lattice ordered Abelian group with strong unit Skeleton Geometric theory 

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References

  1. 1.
    Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Springer Lecture Notes in Mathematics, Berlin (1977)CrossRefzbMATHGoogle Scholar
  2. 2.
    Caramello, O.: Universal models and definability. Math. Proc. Camb. Philos. Soc. 152, 279–302 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Caramello, O.: Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic ‘Bridges’. Oxford University Press, London (2017)zbMATHGoogle Scholar
  4. 4.
    Caramello, O., Russo, A.C.: The Morita-equivalence between MV-algebras and lattice-ordered Abelian groups with strong unit. Journal of Algebra 422, 752–787 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caramello, O., Russo, A.C.: On the geometric theory of local MV-algebras. Journal of Algebra 479, 263–313 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chang, C.C.: A new proof of the completeness of the Łukasiewicz axioms. Trans. Am. Math. Soc. 93, 74–90 (1959)zbMATHGoogle Scholar
  8. 8.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cignoli, R., Dubuc, E.J., Mundici, D.: Extending Stone duality to multisets and locally finite MV-algebras. Journal of Pure and Applied Algebra 189, 37–59 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Di Nola, A., Lettieri, A.: Equational characterization of all varieties of MV-algebras. Journal of Algebra 221, 463–474 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Di Nola, A., Lettieri, A.: One chain generated varieties of MV-algebras. Journal of Algebra 225, 667–697 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Di Nola, A., Esposito, I., Gerla, B.: Local algebras in the representation of MV-algebras. Algebra Universalis 56, 133–164 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Johnstone, P.T.: Sketches of an Elephant: a Topos Theory Compendium, vol. 1-2. Oxford University Press, New York-Oxford (2002)zbMATHGoogle Scholar
  14. 14.
    McLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer, New York (1992)CrossRefGoogle Scholar
  15. 15.
    Mundici, D.: Interpretation of AF C-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mundici, D.: Advanced Łukasiewicz Calculus and MV-Algebras. Springer, Dordrecht (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Antonio Di Nola
    • 1
    • 2
  • Giacomo Lenzi
    • 1
  • Anna Carla Russo
    • 3
  1. 1.University of SalernoFiscianoItaly
  2. 2.I.I.A.S.S. “E. R. Caianiello”Vietri sul mareItaly
  3. 3.PaganiItaly

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