pp 1–21 | Cite as

Some Invariant Skeletons for -u Groups and MV-Algebras

  • Antonio Di Nola
  • Giacomo Lenzi
  • Anna Carla Russo


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).


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Springer Lecture Notes in Mathematics, Berlin (1977)CrossRefMATHGoogle Scholar
  2. 2.
    Caramello, O.: Universal models and definability. Math. Proc. Camb. Philos. Soc. 152, 279–302 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Caramello, O.: Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic ‘Bridges’. Oxford University Press, London (2017)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Caramello, O., Russo, A.C.: On the geometric theory of local MV-algebras. Journal of Algebra 479, 263–313 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chang, C.C.: A new proof of the completeness of the Łukasiewicz axioms. Trans. Am. Math. Soc. 93, 74–90 (1959)MATHGoogle Scholar
  8. 8.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Di Nola, A., Lettieri, A.: Equational characterization of all varieties of MV-algebras. Journal of Algebra 221, 463–474 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Di Nola, A., Lettieri, A.: One chain generated varieties of MV-algebras. Journal of Algebra 225, 667–697 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Di Nola, A., Esposito, I., Gerla, B.: Local algebras in the representation of MV-algebras. Algebra Universalis 56, 133–164 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Johnstone, P.T.: Sketches of an Elephant: a Topos Theory Compendium, vol. 1-2. Oxford University Press, New York-Oxford (2002)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mundici, D.: Advanced Łukasiewicz Calculus and MV-Algebras. Springer, Dordrecht (2011)CrossRefMATHGoogle 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

Personalised recommendations