Opening Reciprocal Relations w.r.t. Stochastic Transitivity

  • Steven Freson
  • Hans De Meyer
  • Bernard De Baets
Part of the Communications in Computer and Information Science book series (CCIS, volume 300)


For crisp as well as fuzzy relations, the results concerning transitive closures and openings are well known. For reciprocal relations transitivity is often defined in terms of stochastic transitivity. This paper focuses on stochastic transitive openings of reciprocal relations, presenting theoretical results as well as a practical method to construct such transitive openings.


Preference relation reciprocal relation stochastic transitivity transitive opening 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bandler, W., Kohout, L.J.: Special properties, closures and interiors of crisp and fuzzy relations. Fuzzy Sets and Systems 26, 317–331 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Basu, K.: Fuzzy revealed preference theory. J. Econom. Theory 32, 212–227 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bezdek, J., Spillman, B., Spillman, R.: A fuzzy relational space for group decision theory. Fuzzy Sets and Systems 1, 255–268 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Boixader, D., Recasens, J.: Transitive openings. In: Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2011) and LFA 2011, pp. 493–497 (2011)Google Scholar
  5. 5.
    Chiclana, F., Herrera-Viedma, E., Alonso, S., Herrera, F.: Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Transactions on Fuzzy Systems 17, 14–23 (2009)CrossRefGoogle Scholar
  6. 6.
    De Baets, B., De Meyer, H.: On the existence and construction of T-transitive closures. Information Sciences 152, 167–179 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    De Baets, B., De Meyer, H.: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems 152, 249–270 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    De Baets, B., De Meyer, H., De Schuymer, B., Jenei, S.: Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare 26, 217–238 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    De Meyer, H., Naessens, H., De Baets, B.: Algorithms for computing the min-transitive closure and associated partition tree of a symmetric fuzzy relation. European Journal of Operational Research 155, 226–238 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Floyd, R.W.: Algorithm 97: shortest path. Communications of the ACM 5(6), 345 (1962)CrossRefGoogle Scholar
  11. 11.
    Freson, S., De Meyer, H., De Baets, B.: An Algorithm for Generating Consistent and Transitive Approximations of Reciprocal Preference Relations. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS, vol. 6178, pp. 564–573. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Freson, S., De Meyer, H., De Baets, B.: On the transitive closure of reciprocal [0,1]-valued relations. In: Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2011) and LFA 2011, pp. 1015–1021 (2011)Google Scholar
  13. 13.
    Monjardet, B.: A generalisation of probabilistic consistency: linearity conditions for valued preference relations. In: Kacprzyk, J., Roubens, M. (eds.) Non-conventional Preference Relations in Decision Making. LNEMS, vol. 301. Springer (1988)Google Scholar
  14. 14.
    Montero, F.J., Tejada, J.: Some problems on the definition of fuzzy preference relations. Fuzzy Sets and Systems 20, 45–53 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Tanino, T.: Fuzzy preference relations in group decision making. LNEMS, vol. 301, pp. 54–71 (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Steven Freson
    • 1
  • Hans De Meyer
    • 1
  • Bernard De Baets
    • 2
  1. 1.Department of Applied Mathematics and Computer ScienceGhent UniversityGentBelgium
  2. 2.Department of Mathematical Modelling, Statistics and BioinformaticsGhent UniversityGentBelgium

Personalised recommendations