Triangular Operations, Negations, and Scalar Cardinality of a Fuzzy Set

  • Maciej Wygralak
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 33)


Scalar approaches to cardinality of a fuzzy set are very simple and convenient, which justifies their frequent use in many areas of applications instead of more advanced and adequate forms such as fuzzy cardinals. On the other hand, theoretical investigations of scalar cardinalities in the hitherto existing subject literature are rather occasional and fragmentary, and lacking in closer references to triangular norms and conorms. This paper is an attempt at filling that gap by constructing an axiomatized theory of scalar cardinality for fuzzy sets with triangular norms and conorms. It brings together all standard scalar approaches, including the so-called sigma-counts and p-powers, and offers infinitely many new alternative options.


Normed Generator Zero Divisor Valuation Property Triangular Norm Fuzzy Coalition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. P. Belluce, A. di Nola and S. Sessa, Triangular norms, MV-algebras and bold fuzzy set theory, Math. Japonica 36(1991)481–487.Google Scholar
  2. [2]
    D. Butnariu and E. P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer Acad. Publ., Dordrecht, 1993.CrossRefGoogle Scholar
  3. [3]
    D. Dubois and H. Prade, A class of fuzzy measures based on tringular norms, Int. J. General Systems 8(1982)43–61.Google Scholar
  4. [4]
    D. Dubois and H. Prade, Fuzzy cardinality and the modeling of imprecise quantification, Fuzzy Sets and Systems 16(1985)199–230.Google Scholar
  5. [5]
    J. C. Fodor, Fuzzy connectives via matrix logic, Fuzzy Sets and Systems 56(1993)67–77.Google Scholar
  6. [6]
    M. J. Frank, On the simultaneous associativity of F(x, y) and x+ y— F(x, y), Aequationes Mathematicae 19(1979)194–226.Google Scholar
  7. [7]
    S. Gottwald, A note on fuzzy cardinals. Kybernetika, 16, 156–158, 1980.MathSciNetMATHGoogle Scholar
  8. [8]
    A. Kaufmann, Introduction à la Théorie des Sous-Ensembles Flous, Vol. IV, Masson, Paris, 1977.Google Scholar
  9. [9]
    M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publ. ( PWN ), Warszawa, 1985.Google Scholar
  10. [10]
    C. H. Ling, Representation of associative functions, Publ. Math. Debrecen 12(1965)189–212.Google Scholar
  11. [11]
    Y. Liu and E. E. Kerre, An overview of fuzzy quantifiers. (I). Interpretation, Fuzzy Sets and Systems 95(1998)1–21.Google Scholar
  12. [12]
    R. Lowen, Fuzzy Set Theory. Basic Concepts, Techniques and Bibliography, Kluwer Acad. Publ., Dordrecht, 1996.Google Scholar
  13. [13]
    A. de Luca and S. Termini, A definition of non-probabilistic entropy in the setting of fuzzy sets theory, Inform. and Control 20(1972)301–312.Google Scholar
  14. [14]
    A. de Luca and S. Termini, Entropy and energy measures of a fuzzy set, in: M. M. Gupta, R. K. Ragade and R. R. Yager, Eds., Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, 1979, 321–338.Google Scholar
  15. [15]
    M. Mizumoto, Pictorial representations of fuzzy connectives. Part I: Cases of t-norms, t-conorms and averaging operators, Fuzzy Sets and Systems 31(1989)217–242.Google Scholar
  16. [16]
    H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, 1997.MATHGoogle Scholar
  17. [17]
    S. Weber, A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems 11(1983)115–134.Google Scholar
  18. [18]
    S. Weber, 1-decomposable measures and integrals for Archimedean t-conorms 1, J. Math. Anal. Appl. 101(1984)114–138.Google Scholar
  19. [19]
    M. Wygralak, Fuzzy cardinals based on the generalized equality of fuzzy subsets, Fuzzy Sets and Systems 18(1986)143–158.Google Scholar
  20. [20]
    M. Wygralak, Generalized cardinal numbers and operations on them, Fuzzy Sets and Systems 53(1993)49–85 (+Erratum, ibid. 62(1994)375).Google Scholar
  21. [21]
    M. Wygralak, Vaguely Defined Objects. Representations, Fuzzy Sets and Nonclassical Cardinality Theory, Kluwer Acad. Publ., Dordrecht, 1996.Google Scholar
  22. [22]
    M. Wygralak, From sigma counts to alternative nonfuzzy cardinalities of fuzzy sets, in: Proc. 7th IPMU Inter. Conf, Paris, 1998, 1339–1344.Google Scholar
  23. [23]
    M. Wygralak, An axiomatic approach to scalar cardinalities of fuzzy sets, Fuzzy Sets and Systems,to appear.Google Scholar
  24. [24]
    M. Wygralak, Questions of cardinality of finite fuzzy sets, Fuzzy Sets and Systems,to appear.Google Scholar
  25. [25]
    L. A. Zadeh, A theory of approximate reasoning, in: J. E. Hayes, D. Michie and L. I. Mikulich, Eds., Machine Intelligence 9, Wiley, New York, 1979, 149–184.Google Scholar
  26. [26]
    L. A. Zadeh, A computational approach to fuzzy quantifiers in natural languages, Comput. and Math. with Appl. 9(1983)149–184.Google Scholar
  27. [27]
    L. A. Zadeh, Knowledge representation in fuzzy logic, IEEE Trans. on Knowledge and Data Engineering 1(1989)89–99.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Maciej Wygralak
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

Personalised recommendations