Exploring Infinitesimal Events through MV-algebras and non-Archimedean States

  • Denisa Diaconescu
  • Anna Rita Ferraioli
  • Tommaso Flaminio
  • Brunella Gerla
Part of the Communications in Computer and Information Science book series (CCIS, volume 443)


In this paper we use tools from the theory of MV-algebras and MV-algebraic states to study infinitesimal perturbations of classical (i.e. Boolean) events and their non-Archimedean probability. In particular we deal with a class of MV-algebras which can be roughly defined by attaching a cloud of infinitesimals to every element of a finite Boolean algebra and for them we introduce the class of Chang-states. These are non-Archimedean mappings which we prove to be representable in terms of a usual (i.e. Archimedean) probability measure and a positive group homomorphism capable to handle the infinitesimal side of the MV-algebras we are dealing with. We also study in which relation Chang-states are with MV-homomorphisms taking value in a suitable perfect MV-algebra.


MV-algebras non-Archimedean states probability measures 


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  1. 1.
    Belluce, P., Di Nola, A., Lettieri, A.: Local MV-algebras. Rend. Circ. Mat. Palermo 42, 347–361 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)Google Scholar
  3. 3.
    Diaconescu, D., Flaminio, T., Leuştean, I.: Lexicographic MV-algebras and lexicographic states (submitted)Google Scholar
  4. 4.
    Di Nola, A., Ferraioli, A.R., Gerla, B.: Combining Boolean algebras and ℓ-groups in the variety generated by Chang’s MV-algebra. Mathematica Slovaca (accepted)Google Scholar
  5. 5.
    Di Nola, A., Georgescu, G., Leustean, I.: States on Perfect MV-algebras. In: Novak, V., Perfilieva, I. (eds.) Discovering the World With Fuzzy Logic. STUDFUZZ, vol. 57, pp. 105–125. Physica, Heidelberg (2000)Google Scholar
  6. 6.
    Di Nola, A., Lettieri, A.: Perfect MV-algebras are categorically equivalent to abelian ℓ-groups. Studia Logica 53, 417–432 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Mundici, D.: Averaging the truth-value in Łukasiewicz logic. Studia Logica 55(1), 113–127 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Mundici, D.: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic, vol. 35. Springer (2011)Google Scholar
  9. 9.
    Paris, J.B.: The uncertain reasoner’s companion: A mathematical perspective. Cambridge University Press (1994)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Denisa Diaconescu
    • 1
  • Anna Rita Ferraioli
    • 2
  • Tommaso Flaminio
    • 2
  • Brunella Gerla
    • 2
  1. 1.Department of Computer Science, Faculty of Mathematics and Computer ScienceUniversity of BucharestRomania
  2. 2.DiSTA - Department of Theoretical and Applied ScienceUniversity of InsubriaItaly

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