Pseudo-Additions and Shift Invariant Aggregation Functions

  • Andrea StupňanováEmail author
  • Doretta Vivona
  • Maria Divari
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 981)


Shift invariant aggregation functions are related to the shifts based on the standard addition \(+\) and their complete characterization is well known. We discuss the aggregation functions invariant with respect to a pseudo-addition \(\oplus \). Our study has two directions. In the first one, we discuss \(\oplus \)-shift invariant aggregation functions with respect to a fixed pseudo-addition \(\oplus \). In the second one, for a fixed aggregation function A, we discuss pseudo-additions \(\oplus \) such that A is \(\oplus \)-shift invariant.


Aggregation function Pseudo-addition Shift invarriantness 


  1. 1.
    Aczél, J., Forte, B., Ng, C.T.: L’équation fonctionnelle triangulaire et la théorie de l’information sans probabilité. C. R. Acad. Sci. Paris Sér. A-B, 275, A727–A729 (1972)Google Scholar
  2. 2.
    Beliakov, G., Bustince Sola, H., Calvo Sánchez, T.: A Practical Guide to Averaging Functions. Springer, Heidelberg (2016)CrossRefGoogle Scholar
  3. 3.
    Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  4. 4.
    Benvenuti, P.: Sulle misure di informazione compositive con traccia compositiva unversale. Rendiconti di Matematica 3–4(2), 481–505 (1969). Serie IVGoogle Scholar
  5. 5.
    Benvenuti, P.: Sur l’independence dans l’information. Colloquies Internationaux du C.N.R.S. Theorie De l’Information, vol. 276, pp. 49–55 (1974)Google Scholar
  6. 6.
    Benvenuti, P., Divari, M., Pandolfi, M.: Su un sistema di equazioni funzionali provenienti dalla teoria soggettiva dell’informazione. Rendiconti di Matematica 39(5), 529–540 (1972). Serie VIzbMATHGoogle Scholar
  7. 7.
    Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R.: Aggregation operators: properties, classes and construction methods. In: Aggregation Operators. Studies in Fuzziness and Soft Computing, vol. 97, pp 3–104. Physica, Heidelberg (2002)Google Scholar
  8. 8.
    Couceiro, M., Marichal, J.-L.: Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices. Fuzzy Sets Syst. 161(5), 694–707 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  10. 10.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  11. 11.
    Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement. Vol. I: Additive and Polynomial Representations. Academic Press, New York (1971)zbMATHGoogle Scholar
  12. 12.
    Lázaro, J., Rückschlossová, T., Calvo, T.: Shift invariant binary aggregation operators. Fuzzy Sets Syst. 142(1), 51–62 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Maslov, V.-P., Samborskii, S.-N.: Idempotent analysis (in place of an introduction). Idempotent Analysis. Advances in Soviet Mathematics, vol. 13, pp. vii–xi. American Mathematical Society, Providence (1992)Google Scholar
  14. 14.
    Mesiar, R., Rückschlossová, T.: Characterization of invariant aggregation operators. Fuzzy Sets Syst. 142(1), 63–73 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mesiar, R., Rybárik, J.: Pseudo-arithmetical operations. Tatra Mount. Math. Publ. 2, 185–192 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mostert, P.-S., Shield, L.: On the structure of semigroups on a compact manifold with boundary. Ann. Math. 65, 117–143 (1957)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pap, E.: Null-additive Set Functions. Mathematics and its Applications, vol. 337. Kluwer Academic Publishers Group, Dordrecht (1995)zbMATHGoogle Scholar
  18. 18.
    Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology (1974)Google Scholar
  19. 19.
    Vivona, D., Divari, M.: Pseudo-analysis: measures of general conditional information. Adv. Sci. Technol. Eng. Syst. J. 2(2), 36–40 (2016)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of Civil EngineeringSlovak University of Technology in BratislavaBratislavaSlovak Republic
  2. 2.Faculty of Civil and Industrial EngineeringSapienza - University of RomeRomeItaly

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