Applied Categorical Structures

, Volume 15, Issue 1–2, pp 135–151 | Cite as

Perfect MV-algebras and their Logic

  • Lawrence P. Belluce
  • Antonio Di Nola
  • Brunella Gerla
Article

Abstract

In this paper, after recounting the basic properties of perfect MV-algebras, we explore the role of such algebras in localization issues. Further, we analyze some logics that are based on Łukasiewicz connectives and are complete with respect to linearly ordered perfect MV-algebras.

Key words

MV-algebras Łukasiewicz logic perfect MV-algebras localization 

Mathematics Subject Classifications (2000)

06D35 03G20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belluce, L.P.: Semisimple algebras of infinite-valued logic. Canad. J. Math. 38, 1356–1379 (1986)MATHMathSciNetGoogle Scholar
  2. 2.
    Belluce, L.P.: The going up and going down theorems in MV-algebras and abelian ℓ-groups. J. Math. Anal. Appl. 241, 92–106 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Belluce, L.P., Chang, C.C.: A weak completeness theorem for infinite valued predicate logic. J. Symbolic Logic 28, 43–50 (1963)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Belluce, L.P., Di Nola, A.: Yosida type representation for perfect MV-algebras. Math. Logic Quart. 42, 551–563 (1996)MATHMathSciNetGoogle Scholar
  5. 5.
    Belluce, L.P., Di Nola, A.: The MV-algebra of first order Łukasiewicz logic. Tatra Mt. Math. Publ. 27, 7–22 (2003)MATHMathSciNetGoogle Scholar
  6. 6.
    Belluce, L.P., Di Nola, A., Lettieri, A.: Local MV-algebras. Rend. Circ. Mat. Palermo 42(2), 347–361 (1993)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Trends in Logic, vol. 7. Kluwer, Dordrecht (2000)Google Scholar
  9. 9.
    Di Nola, A., Lettieri, A.: Perfect MV-algebras are categorically equivalent to abelian ℓ-groups. Stud. Logic 88, 467–490 (1958)Google Scholar
  10. 10.
    Łukasiewicz, J., Tarski, A.: Unntersuchungen über den Aussagenkalkul. Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie 23(cl iii), 30–50 (1930)Google Scholar
  11. 11.
    Mundici, D.: Interpretation of AF C *-algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Scarpellini, B.: Die Nichaxiomatisierbarkeit des unendlichwertigen Pradikatenkalkulus von Lukasiewicz. J. Symbolic Logic 27, 159–170 (1962)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Lawrence P. Belluce
    • 1
  • Antonio Di Nola
    • 2
  • Brunella Gerla
    • 3
  1. 1.Department of MathematicsBritish Columbia UniversityVancouverCanada
  2. 2.Department of Mathematics and InformaticsUniversity of SalernoBaronissiItaly
  3. 3.Department of Mathematics and InformaticsUniversity of InsubriaVareseItaly

Personalised recommendations