Advertisement

Transitivity and Negative Transitivity in the Fuzzy Setting

  • Susana Díaz
  • Bernard De Baets
  • Susana Montes
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 107)

Abstract

A (crisp) binary relation is transitive if and only if its dual relation is negatively transitive. In preference modelling, if a weak preference relation is complete, the associated strict preference relation is its dual relation. It follows from here this well-known result: given a complete weak preference relation, it is transitive if and only if its strict preference relation is negatively transitive.

In the context of fuzzy relations, transitivity is traditionally defined by a t-norm and negative transitivity, by a t-conorm. In this setting, it is also well known that a (valued) binary relation is T-transitive if and only if its dual relation is negatively S-transitive where S stands for the dual t-conorm of the t-norm T. However, in this context there are several proposals to get the strict preference relation from the weak preference relation. Also, there are different definitions of completeness. In this contribution we depart from a reflexive fuzzy relation. We assume that this relation is transitive with respect to a conjunctor (a generalization of t-norms). We consider almost all the possible generators and therefore all the possible strict preference relations obtained from the reflexive relation and we provide a general expression for the negative transitivity that those relations satisfy.

Keywords

Preference Structure Fuzzy Relation Dual Relation Fuzzy Preference Rela Indifference Relation 
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.
    Arrow, K.J.: Social Choice and Individual Values. Wiley, Chichester (1951)zbMATHGoogle Scholar
  2. 2.
    Bodenhofer, U., Klawonn, F.: A formal study of linearity axioms for fuzzy orderings. Fuzzy Sets and Systems 145, 323–354 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    De Baets, B., Fodor, J.: Twenty years of fuzzy preference structures (1978-1997. Belg. J. Oper. Res. Statist. Comput. Sci. 37, 61–82 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    De Baets, B., Fodor, J.: Additive fuzzy preference structures: the next generation. In: De Baets, B., Fodor, J. (eds.) Principles of Fuzzy Preference Modelling and Decision Making, pp. 15–25. Academic Press, London (2003)Google Scholar
  5. 5.
    De Baets, B., Van de Walle, B.: Minimal definitions of classical and fuzzy preference structures. In: Proceedings of the Annual Meeting of the North American Fuzzy Information Processing Society, USA, Syracuse, New York, pp. 299–304 (1997)Google Scholar
  6. 6.
    Díaz, S., De Baets, B., Montes, S.: Additive decomposition of fuzzy pre-orders. Fuzzy Sets and Systems 158, 830–842 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Díaz, S., De Baets, B., Montes, S.: On the compositional characterization of complete fuzzy pre-orders. Fuzzy Sets and Systems 159, 2221–2239 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Díaz, S., De Baets, B., Montes, S.: General results on the decomposition of transitive fuzzy relations. Fzzy Optim. Decis. Making 9, 1–29 (2010)zbMATHCrossRefGoogle Scholar
  9. 9.
    Díaz, S., Montes, S., De Baets, B.: Transitive decomposition of fuzzy preference relations: the case of nilpotent minimum. Kybernetika 40, 71–88 (2004)MathSciNetGoogle Scholar
  10. 10.
    Díaz, S., Montes, S., De Baets, B.: Transitivity bounds in additive fuzzy preference structures. IEEE Trans. on Fuzzy Systems 15, 275–286 (2007)CrossRefGoogle Scholar
  11. 11.
    Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)zbMATHGoogle Scholar
  12. 12.
    Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)zbMATHGoogle Scholar
  13. 13.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)zbMATHGoogle Scholar
  14. 14.
    Roubens, M., Vincke, P.: Preference Modelling. Lecture Notes in Economics and Mathematical Systems, vol. 76. Springer, Heidelberg (1998)Google Scholar
  15. 15.
    Van de Walle, B., De Baets, B., Kerre, E.: A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. Part 1: General argumentation. Fuzzy Sets and Systems 97, 349–359 (1998)zbMATHGoogle Scholar
  16. 16.
    Van de Walle, B., De Baets, B., Kerre, E.: Characterizable fuzzy preference structures. Annals of Operations Research 80, 105–136 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Susana Díaz
    • 1
  • Bernard De Baets
    • 2
  • Susana Montes
    • 1
  1. 1.Dept. Statistics and O. R.University of OviedoSpain
  2. 2.Dept. Appl. Math., Biometrics and Process ControlGhent UniversityBelgium

Personalised recommendations