Qualitative Integrals and Cointegrals: A Survey

  • Agnès RicoEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 981)


Qualitatives integrals and cointegrals are aggregation functions defined on qualitative scales where only minimum, maximum and order reversing map are allowed. Their definition uses a conjunction or an implication linked by property of semi-duality. They generalise Sugeno integrals which are defined using the Kleene-Dienes implication or conjunction. This survey remembers the definitions, the properties and some characterisation theorems for these qualitative (co)integrals.


  1. 1.
    Dubois, D., Rico, A., Prade, H., Teheux, B.: Characterizing variants of qualitative Sugeno integrals in a totally ordered Heyting algebra. In: Proceedings of 9th Conference of the European Society for Fuzzy Logic and Technology (Eusflat), Gijon, pp. 862–872 (2015)Google Scholar
  2. 2.
    Fodor, J.: On fuzzy implication operators. Fuzzy Sets Syst. 42, 293–300 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Weighted minimum and maximum operations. Inf. Sci. 39, 205–210 (1986)CrossRefGoogle Scholar
  4. 4.
    Dubois, D., Prade, H., Rico, A.: Residuated variants of Sugeno integrals. Inf. Sci. 329, 765–781 (2016)CrossRefGoogle Scholar
  5. 5.
    Sugeno, M.: Theory of Fuzzy Integrals and its Applications. Ph.D thesis, Tokyo Institute of Technology (1974)Google Scholar
  6. 6.
    Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy measure of fuzzy events defined by fuzzy integrals. Fuzzy Sets Syst. 50(3), 293–313 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borzová-Molnárová, J., Halčinová, L., Hutník, O.: The smallest semicopula-based universal integrals I: properties and characterizations. Fuzzy Sets Syst. 271, 1–15 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dvořák, A., Holčapek, M.: Fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices. Inf. Sci. 185, 205–229 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaufman, A., Le calcul des admissibilités: Une idée nouvelle à partir de la théorie des sous-ensembles flous. In: Proceedings of Colloque International sur la Théorie et les Applications des Sous-Ensembles Flous, vol. I, p. 14, Marseilles (1978)Google Scholar
  10. 10.
    Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dubois, D., Prade, H., Rico, A., Teheux, B.: Generalized qualitative Sugeno integrals. Inf. Sci. 415, 429–445 (2017)CrossRefGoogle Scholar
  12. 12.
    Agahi, H., Mesiar, R., Ouyang, Y.: New general extensions of Chebyshev type inequalities for Sugeno integrals. Int. J. Approx. Reason. 51, 135–140 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Agahi, H., Mesiar, R., Ouyang, Y.: General Minkowski type inequalities for Sugeno integrals. Fuzzy Sets Syst. 161, 708–715 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ouyang, Y., Mesiar, R., Agahi, H.: An inequality related to Minkowski type for Sugeno integrals. Inf. Sci. 180, 2793–2801 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yan, T., Ouyang, Y.: Chebyshev inequality for q-integrals. Int. J. Approx. Reason. 106, 146–154 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Flores-Franulič, A., Román-Flores, H.: A Chebyshev type inequality for fuzzy integrals. Appl. Math. Comput. 190, 1178–1184 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mesiar, R., Ouyang, Y.: General Chebyshev type inequalities for Sugeno integrals. Fuzzy Sets Syst. 160, 58–64 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Klement, E.P., Mesiar, R., Pap, E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Syst. 18, 178–187 (2010)CrossRefGoogle Scholar
  19. 19.
    Boczek, M., Kaluszka, M.: On conditions under which some generalized Sugeno integrals coincide: a solution to Dubois’ problem. Fuzzy Sets Syst. 326, 81–88 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sugeno, M.: Fuzzy measures and fuzzy integrals: a survey. In: Gupta, M.M., et al. (eds.) Fuzzy Automata and Decision Processes, North-Holland, pp. 89–102 (1977)Google Scholar
  21. 21.
    Dubois, D., Prade, H., Sabbadin, R.: Qualitative decision theory with Sugeno integrals. In: Grabisch, M., et al. (eds.) Fuzzy Measures and Integrals Theory and Applications, pp. 314–322. Physica Vrlg, Heidelberg (2000)zbMATHGoogle Scholar
  22. 22.
    Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann. Oper. Res. 175, 247–286 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Dubois, D.: Fuzzy measures on finite scales as families of possibility measures. In: Proceedings of EUSFLAT-LFA Conference, Aix-Les-Bains, France pp. 822–829 (2011)Google Scholar
  24. 24.
    Banon, G.: Constructive decomposition of fuzzy measures in terms of possibility and necessity measures. In: Proceedings of VIth IFSA World Congress, São Paulo, Brazil, vol. I, p. 217–220 (1995)Google Scholar
  25. 25.
    Dubois, D., Prade, H., Rico, A.: Qualitative capacities as imprecise possibilities. In: Van der Gaag, L. (ed.) Symbolic and Quantitative Approaches to Reasoning with Uncertainty, (ECSQARU 2013). Lecture Notes in Computer Science, vol. 7958, pp. 169–180. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  26. 26.
    Dubois, D., Prade, H.: A theorem on implication functions defined from triangular norms. Stochastica 8, 267–279 (1984)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.ERIC & Univ. Claude Bernard Lyon 1VilleurbanneFrance

Personalised recommendations