Advertisement

The Dominance Relation on the Class of Continuous T-Norms from an Ordinal Sum Point of View

  • Susanne Saminger
  • Peter Sarkoci
  • Bernard De Baets
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4342)

Abstract

This paper addresses the relation of dominance on the class of continuous t-norms with a particular focus on continuous ordinal sum t-norms. Exactly, in this framework counter-examples to the conjecture that dominance is not only a reflexive and antisymmetric, but also a transitive relation could be found. We elaborate the details which have led to these results and illustrate them by several examples. In addition, to this original and comprehensive overview, we provide geometrical insight into dominance relationships involving prototypical Archimedean t-norms, the Łukasiewicz t-norm and the product t-norm.

Keywords

Dominance Relation Dominance Relationship Idempotent Element Triangular Norm Order Isomorphism 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alsina, C., Frank, M., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Company, Singapore (2006)MATHCrossRefGoogle Scholar
  2. 2.
    Bodenhofer, U.: Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems 137(1), 113–136Google Scholar
  3. 3.
    Bodenhofer, U.: A Similarity-Based Generalization of Fuzzy Orderings. Schriftenreihe der Johannes-Kepler-Universität Linz, vol. C 26. Universitätsverlag Rudolf Trauner (1999)Google Scholar
  4. 4.
    Butnariu, D., Klement, E.P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Theory and Decision Library, Series C: Game Theory, Mathematical Programming and Operations Research, vol. 10. Kluwer Academic Publishers, Dordrecht (1993)MATHGoogle Scholar
  5. 5.
    Clifford, A.H.: Naturally totally ordered commutative semigroups. Amer. J. Math. 76, 631–646 (1954)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Clifford, A.H., Preston, G.B.: The algebraic theory of semigroups. American Mathematical Society, Providence (1961)Google Scholar
  7. 7.
    Climescu, A.C.: Sur l’équation fonctionelle de l’associativité. Bull. École Polytechn. Iassy 1, 1–16 (1946)Google Scholar
  8. 8.
    De Baets, B., Mesiar, R.: T-partitions. Fuzzy Sets and Systems 97, 211–223 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Díaz, S., Montes, S., De Baets, B.: Transitivity bounds in additive fuzzy preference structures. IEEE Trans. Fuzzy Systems (in press, 2006)Google Scholar
  10. 10.
    Dubois, D., Prade, H.: New results about properties and semantics of fuzzy set-theoretic operators. In: Wang, P.P., Chang, S.K. (eds.) Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, pp. 59–75. Plenum Press, New York (1980)Google Scholar
  11. 11.
    Fodor, J.C., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)MATHGoogle Scholar
  12. 12.
    Frank, M.J.: On the simultaneous associativity of F(x,y) and x + y − F(x,y). Aequationes Mathematicae 19, 194–226 (1979)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Grabisch, M., Nguyen, H.T., Walker, E.A.: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  14. 14.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  15. 15.
    Hofmann, K.H., Lawson, J.D.: Linearly ordered semigroups: Historic origins and A. H. Clifford’s influence. London Math. Soc. Lecture Notes, vol. 231, pp. 15–39. Cambridge University Press, Cambridge (1996)Google Scholar
  16. 16.
    Klein-Barmen, F.: Über gewisse Halbverbände und kommutative Semigruppen II. Math. Z. 48, 715–734 (1942-1943)Google Scholar
  17. 17.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. In: Trends in Logic. Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  18. 18.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy Sets and Systems 143, 5–26 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper II: general constructions and parameterized families. Fuzzy Sets and Systems 145, 411–438 (2004)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper III: continuous t-norms. Fuzzy Sets and Systems 145, 439–454 (2004)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Klement, E.P., Weber, S.: Generalized measures. Fuzzy Sets and Systems 40, 375–394 (1991)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ling, C.M.: Representation of associative functions. Publ. Math. Debrecen 12, 189–212 (1965)MathSciNetGoogle Scholar
  23. 23.
    Mayor, G., Torrens, J.: On a family of t-norms. Fuzzy Sets and Systems 41, 161–166 (1991)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. U.S.A. 8, 535–537 (1942)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Mostert, P.S., Shields, A.L.: On the structure of semi-groups on a compact manifold with boundary. Ann. of Math. II. Ser. 65, 117–143 (1957)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Nelsen, R.B.: An Introduction to Copulas. Lecture Notes in Statistics, vol. 139. Springer, New York (1999)MATHGoogle Scholar
  27. 27.
    Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht (1995)MATHGoogle Scholar
  28. 28.
    Preston, G.B.: A. H. Clifford: an appreciation of his work on the occasion of his sixty-fifth birthday. Semigroup Forum 7, 32–57 (1974)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Saminger, S.: Aggregation in Evaluation of Computer-Assisted Assessment. In: Schriftenreihe der Johannes-Kepler-Universität Linz, vol. C 44, Universitätsverlag Rudolf Trauner (2005)Google Scholar
  30. 30.
    Saminger, S., De Baets, B., De Meyer, H.: On the dominance relation between ordinal sums of conjunctors. Kybernetika 42(3), 337–350 (2006)MathSciNetGoogle Scholar
  31. 31.
    Saminger, S., De Baets, B., De Meyer, H.: The domination relation between continuous t-norms. In: Proceedings of Joint 4th Int. Conf. in Fuzzy Logic and Technology and 11th French Days on Fuzzy Logic and Applications, Barcelona, Spain, pp. 247–252 (September 2005)Google Scholar
  32. 32.
    Saminger, S., Mesiar, R., Bodenhofer, U.: Domination of aggregation operators and preservation of transitivity. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10/s, 11–35 (2002)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Sarkoci, P.: Dominance of ordinal sums of Łukasiewicz and product t-norm (forthcoming)Google Scholar
  34. 34.
    Sarkoci, P.: Domination in the families of Frank and Hamacher t-norms. Kybernetika 41, 345–356 (2005)MathSciNetGoogle Scholar
  35. 35.
    Sarkoci, P.: Dominance is not transitive on continuous triangular norms. Aequationes Mathematicae (submitted, 2006)Google Scholar
  36. 36.
    Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960)MATHMathSciNetGoogle Scholar
  37. 37.
    Schweizer, B., Sklar, A.: Associative functions and abstract semigroups. Publ. Math. Debrecen 10, 69–81 (1963)MathSciNetGoogle Scholar
  38. 38.
    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983)MATHGoogle Scholar
  39. 39.
    Šerstnev, A.N.: Random normed spaces: problems of completeness. Kazan. Gos. Univ. Učen. Zap. 122, 3–20 (1962)Google Scholar
  40. 40.
    Sherwood, H.: Characterizing dominates on a family of triangular norms. Aequationes Mathematicae 27, 255–273 (1984)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Sugeno, M.: Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology (1974)Google Scholar
  42. 42.
    Tardiff, R.M.: Topologies for probabilistic metric spaces. Pacific J. Math. 65, 233–251 (1976)MATHMathSciNetGoogle Scholar
  43. 43.
    Tardiff, R.M.: On a functional inequality arising in the construction of the product of several metric spaces. Aequationes Mathematicae 20, 51–58 (1980)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Tardiff, R.M.: On a generalized Minkowski inequality and its relation to dominates for t-norms. Aequationes Mathematicae 27, 308–316 (1984)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Weber, S.: ⊥-decomposable measures and integrals for Archimedean t-conorms ⊥. J. Math. Anal. Appl. 101, 114–138 (1984)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Zadeh, L.A.: Fuzzy sets. Inform. and Control 8, 338–353 (1965)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Susanne Saminger
    • 1
  • Peter Sarkoci
    • 2
  • Bernard De Baets
    • 3
  1. 1.Department of Knowledge-Based Mathematical SystemsJohannes Kepler UniversityLinzAustria
  2. 2.Department of MathematicsIIEAM, Slovak University of TechnologyBratislavaSlovakia
  3. 3.Department of Applied Mathematics, Biometrics, and Process ControlGhent UniversityGentBelgium

Personalised recommendations