OM3: Ordered Maxitive, Minitive, and Modular Aggregation Operators: Axiomatic Analysis under Arity-Dependence (I)

  • Anna Cena
  • Marek Gagolewski
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 228)


Recently, a very interesting relation between symmetric minitive, maxitive, and modular aggregation operators has been shown. It turns out that the intersection between any pair of the mentioned classes is the same. This result introduces what we here propose to call the OM3 operators. In the first part of our contribution on the analysis of the OM3 operators we study some properties that may be useful when aggregating input vectors of varying lengths. In Part II we will perform a thorough simulation study of the impact of input vectors’ calibration on the aggregation results.


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  1. 1.
    Beliakov, G., James, S.: Stability of weighted penalty-based aggregation functions. Fuzzy Sets and Systems (2013), doi:10.1016/j.fss.2013.01.007Google Scholar
  2. 2.
    Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. STUDFUZZ, vol. 221. Springer, Heidelberg (2007)Google Scholar
  3. 3.
    Calvo, T., Mayor, G., Torrens, J., Suner, J., Mas, M., Carbonell, M.: Generation of weighting triangles associated with aggregation functions. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 8(4), 417–451 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Calvo, T., Kolesarova, A., Komornikova, M., Mesiar, R.: Aggregation operators: Properties, classes and construction methods. In: Calvo, T., Mayor, G., Mesiar, R. (eds.) Aggregation Operators. New Trends and Applications. STUDFUZZ, vol. 97, pp. 3–104. Physica-Verlag, New York (2002)CrossRefGoogle Scholar
  5. 5.
    Dubois, D., Prade, H., Testemale, C.: Weighted fuzzy pattern matching. Fuzzy Sets and Systems 28, 313–331 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Franceschini, F., Maisano, D.A.: The Hirsch index in manufacturing and quality engineering. Quality and Reliability Engineering International 25, 987–995 (2009)CrossRefGoogle Scholar
  7. 7.
    Gagolewski, M.: On the relation between effort-dominating and symmetric minitive aggregation operators. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part III. CCIS, vol. 299, pp. 276–285. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Gagolewski, M.: On the relationship between symmetric maxitive, minitive, and modular aggregation operators. Information Sciences 221, 170–180 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gągolewski, M., Grzegorzewski, P.: Arity-monotonic extended aggregation operators. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 80, pp. 693–702. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Gagolewski, M., Grzegorzewski, P.: Axiomatic characterizations of (quasi-) L-statistics and S-statistics and the Producer Assessment Problem. In: Galichet, S., Montero, J., Mauris, G. (eds.) Proc. Eusflat/LFA 2011, pp. 53–58 (2011)Google Scholar
  11. 11.
    Gagolewski, M., Grzegorzewski, P.: Possibilistic analysis of arity-monotonic aggregation operators and its relation to bibliometric impact assessment of individuals. International Journal of Approximate Reasoning 52(9), 1312–1324 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gagolewski, M., Mesiar, R.: Aggregating different paper quality measures with a generalized h-index. Journal of Informetrics 6(4), 566–579 (2012)CrossRefGoogle Scholar
  13. 13.
    Ghiselli Ricci, R., Mesiar, R.: Multi-attribute aggregation operators. Fuzzy Sets and Systems 181(1), 1–13 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation functions, Cambridge (2009)Google Scholar
  15. 15.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes. Information Sciences 181, 23–43 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation functions: Means. Information Sciences 181, 1–22 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hirsch, J.E.: An index to quantify individual’s scientific research output. Proceedings of the National Academy of Sciences 102(46), 16,569–16,572 (2005)Google Scholar
  18. 18.
    Klement, E., Manzi, M., Mesiar, R.: Ultramodular aggregation functions. Information Sciences 181, 4101–4111 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Mayor, G., Calvo, T.: On extended aggregation functions. In: Proc. IFSA 1997, vol. 1, pp. 281–285. Academia, Prague (1997)Google Scholar
  20. 20.
    Mesiar, R., Mesiarová-Zemánková, A.: The ordered modular averages. IEEE Transactions on Fuzzy Systems 19(1), 42–50 (2011)CrossRefGoogle Scholar
  21. 21.
    Mesiar, R., Pap, E.: Aggregation of infinite sequences. Information Sciences 178, 3557–3564 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Woeginger, G.J.: An axiomatic analysis of Egghe’s g-index. Journal of Informetrics 2(4), 364–368 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Woeginger, G.J.: An axiomatic characterization of the Hirsch-index. Mathematical Social Sciences 56(2), 224–232 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Yager, R., Rybalov, A.: Noncommutative self-identity aggregation. Fuzzy Sets and Systems 85, 73–82 (1997)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Systems Research InstitutePolish Academy of SciencesWarsawPoland
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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