Sugeno integrals are aggregation functions defined on a qualitative scale where only minimum, maximum and order-reversing maps are allowed. Recently, variants of Sugeno integrals based on Gödel implication and its contraposition were defined and axiomatized in the setting of bounded chain with an involutive negation. This paper proposes a more general approach. We consider totally ordered scales, multivalued conjunction operations not necessarily commutative, and implication operations induced from them by means of an involutive negation. In such a context, different Sugeno-like integrals are defined and axiomatized.


Sugeno integral Conjunctions Implications Multifactorial evaluation 



This work is partially supported by ANR-11-LABX-0040-CIMI (Centre International de Mathématiques et d’Informatique) within the program ANR-11-IDEX-0002-02, project ISIPA.


  1. 1.
    Chateauneuf, A., Grabisch, M., Rico, A.: Modeling attitudes toward uncertainty through the use of the Sugeno integral. J. Math. Econ. 44, 1084–1099 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dubois, D., Prade, H.: A theorem on implication functions defined from triangular norms. Stochastica 8, 267–279 (1984)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New-York (1988)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dubois, D., Prade, H., Rico, A.: Residuated variants of Sugeno integrals. Inf. Sci. 329, 765–781 (2016)CrossRefGoogle Scholar
  5. 5.
    Dubois, D., Rico, A., Teheux, B., Prade, H.: 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. 865–872 (2015)Google Scholar
  6. 6.
    Dvořák, A., Holčapek, M.: Fuzzy integrals over complete residuated lattices. In: Proceedings of the IFSA-EUSFLAT Conference, Lisbon, pp. 357–362 (2009)Google Scholar
  7. 7.
    Dvořák, A., Holčapek, M.: Fuzzy measures, integrals defined on algebras of fuzzy subsets over complete residuated lattices. Inf. Sci. 185, 205–229 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fodor, J.: On fuzzy implication operators. Fuzzy Sets Syst. 42, 293–300 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Marichal, J.-L.: On Sugeno integrals as an aggregation function. Fuzzy Sets Syst. 114, 347–365 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Klement, E., 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
  11. 11.
    Sugeno, M.: Fuzzy measures, fuzzy integrals: A survey. In: Gupta, M.M., et al. (eds.) Fuzzy Automata and Decision Processes, pp. 89–102. North-Holland, Amsterdam (1977)Google Scholar
  12. 12.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Didier Dubois
    • 1
    Email author
  • Henri Prade
    • 1
  • Agnès Rico
    • 2
  • Bruno Teheux
    • 3
  1. 1.IRIT, Université Paul SabatierToulouse Cedex 9France
  2. 2.ERICUniversité Claude Bernard Lyon 1VilleurbanneFrance
  3. 3.Mathematics Research Unit, FSTCUniversity of LuxembourgLuxembourgLuxembourg

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