On k–additive Aggregation Functions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9880)


Inspired by the Grabisch idea of k–additive measures, we introduce and study k–additive aggregation functions. The Owen multilinear extension of a k–additive capacity is shown to be a particular k–additive aggregation function. We clarify the relation between k–additive aggregation functions and polynomials of a degree not exceeding k. We also describe \(n^2 + 2n\) basic 2–additive n–ary aggregation functions whose convex closure forms the class of all 2–additive n–ary aggregation functions.


Aggregation function k–additive aggregation function k–additive capacity 


  1. 1.
    Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg (2007)MATHGoogle Scholar
  2. 2.
    Beliakov, G., Bustince, H., Calvo, T.: A Practical Guide to Averaging Functions. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  3. 3.
    Chateauneuf, A., Jaffray, J.Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Math. Soc. Sci. 17, 263–283 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Grabisch, M.: \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92, 167–189 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Kolesárová, A., Stupňanová, A., Beganová, J.: Aggregation-based extensions of fuzzy measures. Fuzzy Sets Syst. 194, 1–14 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lovász, L.: Submodular function and convexity. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming: The state of the art, pp. 235–257. Springer, Berlin (1983)CrossRefGoogle Scholar
  9. 9.
    Marichal, J.-L.: Aggregation of interacting criteria by means of the discrete Choquet integral. In: Calvo, T., Mayor, G., Mesiar, R. (eds.) Aggregation Operators. New Trends and Applications, pp. 224–24. Physica-Verlag, Heidelberg (2002)Google Scholar
  10. 10.
    Owen, G.: Multilinear extensions of games. In: Shapley, S., Roth, A.E. (eds.) The Shapley value. Essays in Honour of Lloyd, pp. 139–151. Cambridge University Press, Cambridge (1988)Google Scholar
  11. 11.
    Valášková, L.: A note to the 2-order additivity. In: Proceedings of MAGIA, Kočovce, pp. 53–55 (2001)Google Scholar
  12. 12.
    Valášková, L.: Non-additive measures and integrals. Ph.D. thesis, STU Bratislava, (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Chemical and Food TechnologySlovak University of TechnologyBratislavaSlovakia
  2. 2.School of ScienceCommunication University of ChinaBeijingPeople’s Republic of China
  3. 3.Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovakia

Personalised recommendations