On States on Fuzzy MV-algebra

  • Alžbeta MichalíkováEmail author
  • Beloslav Riečan
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1081)


Some MV-algebras generated by families of fuzzy sets are considered. On the fuzzy MV-algebras a representation theorem is presented as well as a theorem of an invariant state in the case that the basic set is a locally compact topological group.


Locally compact topological groups Invariant states Intuitionistic fuzzy sets MV-algebras 


  1. 1.
    Atanassov, K.T.: Intuitionistic Fuzzy Sets: Theory and Applications. Physic Verlag, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Butnariu, D., Klement, E.P.: Triangular norm-based measures. In: Triangular Norm-Based Measures and Games with Fuzzy Coalitions, pp. 37–68. Springer, Dordrecht (1993)Google Scholar
  3. 3.
    Ciungu, L., Riečan, B.: General form of probabilities on IF-sets. In: Fuzzy Logic and Applications, pp. 101–107 (2009)Google Scholar
  4. 4.
    Ciungu, L.C., Riečan, B.: Representation theorem for probabilities on IFS-events. Inf. Sci. 180(5), 793–798 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Čunderlíková, K., Riečan, B.: On two formulations of the representation theorem for an IF-state. In: International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets, pp. 65–70. Springer, Cham (2016)Google Scholar
  6. 6.
    Grzegorzewski, P., Mrówka, E.: Probability of intuitionistic fuzzy events. In: Soft Methods in Probability, Statistics and Data Analysis, pp. 105–115. Physica, Heidelberg (2002)Google Scholar
  7. 7.
    Halmos, P.R.: Measure Theory. Springer, New York (1950) CrossRefGoogle Scholar
  8. 8.
    Jurečková, M., Riečan, B.: Weak law of large numbers for weak observables in MV algebras. Tatra Mt. Math. Publ. 12, 221–228 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Michalíková, A., Riečan, B.: On invariant measures on intuitionistic fuzzy sets. In: Advances in Fuzzy Logic and Technology 2017, pp. 529–534. Springer, Cham (2017)Google Scholar
  10. 10.
    Riečan, B.: On the conditional expectation of observables in MV algebras of fuzzy sets. Fuzzy Sets Syst. 102(3), 445–450 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Riečan, B.: On the probability theory on MV algebras. Soft Comput. Fusion Found. Methodol. Appl. 4(1), 49–57 (2000). A Fusion of Foundations, Methodologies and ApplicationszbMATHGoogle Scholar
  12. 12.
    Riečan, B.: Kolmogorov - Sinaj entropy on MV-algebras. Int. J. Theor. Phys. 44(7), 1041–1052 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Riečan, B.: On a problem of Radko Mesiar: general form of IF-probabilities. Fuzzy Sets Syst. 157(11), 1485–1490 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Riečan, B.: Analysis of fuzzy logic models. In: Koleshko, V. (ed.) Intelligent Systems, INTECH, pp. 217–244 (2012)Google Scholar
  15. 15.
    Riečan, B.: Strong Poincaré recurrence theorem in MV-algebras. Math. Slovaca 60(5), 655–664 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Riečan, B., Lašová, L.: On the probability theory on the Kôpka D-posets. In: Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, vol. 1, pp. 167–176 (2010)Google Scholar
  17. 17.
    Riečan, B., Mundici, D.: Probability on MV-algebras. Handb. Meas. Theory 21, 869–910 (2002). E. Pap Ed. Elsevier Science, AmsterdamMathSciNetzbMATHGoogle Scholar
  18. 18.
    Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering, vol. 411. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  19. 19.
    Zadeh, L.A.: Information and control. Fuzzy Sets 8(3), 338–353 (1965)Google Scholar
  20. 20.
    Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23(2), 421–427 (1968)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning I. Inf. Sci. 8(3), 199–249 (1975)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of Natural ScienceMatej Bel UniversityBanská BystricaSlovakia
  2. 2.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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