General Representation Theorems for Fuzzy Weak Orders

  • Ulrich Bodenhofer
  • Bernard De Baets
  • János Fodor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4342)


The present paper gives a state-of-the-art overview of general representation results for fuzzy weak orders. We do not assume that the underlying domain of alternatives is finite. Instead, we concentrate on results that hold in the most general case that the underlying domain is possibly infinite. This paper presents three fundamental representation results: (i) score function-based representations, (ii) inclusion-based representations, (iii) representations by decomposition into crisp linear orders and fuzzy equivalence relations.


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  1. 1.
    Bandler, W., Kohout, L.J.: Fuzzy power sets and fuzzy implication operators. Fuzzy Sets and Systems 4, 183–190 (1980)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bandler, W., Kohout, L.J.: Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems. In: Wang, S.K., Chang, P.P. (eds.) Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, pp. 341–367. Plenum Press, New York (1980)Google Scholar
  3. 3.
    Bodenhofer, U.: Representations and constructions of strongly linear fuzzy orderings. In: Proc. EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, pp. 215–218 (September 1999)Google Scholar
  4. 4.
    Bodenhofer, U.: A similarity-based generalization of fuzzy orderings preserving the classical axioms. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 8(5), 593–610 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bodenhofer, U.: Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems 137(1), 113–136 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bodenhofer, U., Küng, J.: Fuzzy orderings in flexible query answering systems. Soft Computing 8(7), 512–522 (2004)MATHCrossRefGoogle Scholar
  7. 7.
    Cantor, G.: Beiträge zur Begründung der transfiniten Mengenlehre. Math. Ann. 46, 481–512 (1895)CrossRefGoogle Scholar
  8. 8.
    De Baets, B., Fodor, J., Kerre, E.E.: Gödel representable fuzzy weak orders. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 7(2), 135–154 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    De Baets, B., Mesiar, R.: Pseudo-metrics and T-equivalences. J. Fuzzy Math. 5(2), 471–481 (1997)MATHMathSciNetGoogle Scholar
  10. 10.
    De Baets, B., Mesiar, R.: Metrics and T-equalities. J. Math. Anal. Appl. 267, 331–347 (2002)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Fodor, J., Ovchinnikov, S.V.: On aggregation of T-transitive fuzzy binary relations. Fuzzy Sets and Systems 72, 135–145 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)MATHGoogle Scholar
  13. 13.
    Gottwald, S.: Fuzzy Sets and Fuzzy Logic. Vieweg, Braunschweig (1993)Google Scholar
  14. 14.
    Gottwald, S.: A Treatise on Many-Valued Logics. In: Studies in Logic and Computation, Research Studies Press, Baldock (2001)Google Scholar
  15. 15.
    Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  16. 16.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)MATHGoogle Scholar
  17. 17.
    Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement. Academic Press, San Diego (1971)MATHGoogle Scholar
  18. 18.
    Ovchinnikov, S.V.: Similarity relations, fuzzy partitions, and fuzzy orderings. Fuzzy Sets and Systems 40(1), 107–126 (1991)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ovchinnikov, S.V.: An introduction to fuzzy relations. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets, vol. 7, pp. 233–259. Kluwer Academic Publishers, Boston (2000)Google Scholar
  20. 20.
    Ovchinnikov, S.V.: Numerical representation of transitive fuzzy relations. Fuzzy Sets and Systems 126, 225–232 (2002)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Roberts, F.S.: Measurement Theory. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  22. 22.
    Rosenstein, J.G.: Linear Orderings. Pure and Applied Mathematics, vol. 98. Academic Press, New York (1982)MATHGoogle Scholar
  23. 23.
    Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fund. Math. 16, 386–389 (1930)MATHGoogle Scholar
  24. 24.
    Valverde, L.: On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17(3), 313–328 (1985)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Inform. Sci. 3, 177–200 (1971)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ulrich Bodenhofer
    • 1
  • Bernard De Baets
    • 2
  • János Fodor
    • 3
  1. 1.Institute of BioinformaticsJohannes Kepler UniversityLinzAustria
  2. 2.Dept. of Applied Mathematics, Biometrics, and Process ControlGhent UniversityGentBelgium
  3. 3.Institute of Intelligent Engineering SystemsBudapest TechBudapestHungary

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