Advertisement

The Relation Between F-partial Order and Distributivity Equation

  • Emel AşıcıEmail author
  • Radko Mesiar
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 981)

Abstract

The notations of the order induced by triangular norms, nullnorms and uninorms have been studied widely. Nullnorms have been produced from triangular norms and triangular conorms with a zero element in the interior of the unit interval. They have been proved to be useful in several fields like expert systems, neural networks, fuzzy quantifiers. Also, the distributivity equation has been studied involving different classes of aggregation functions from triangular norms and triangular conorms to nullnorms. So, the study of the distributive property becomes very interesting, since nullnorms have been used several fields. In this paper we investigate distributivity equation for nullnorms on the unit interval [0, 1] and we give sufficient condition for two nullnorms to be equivalent.

Notes

Acknowledgement

Second author was supported of grants VEGA 1/0006/19.

References

  1. 1.
    Aşıcı, E.: On the properties of the \(F\)-partial order and the equivalence of nullnorms. Fuzzy Sets Syst. 346, 72–84 (2018)Google Scholar
  2. 2.
    Aşıcı, E.: An order induced by nullnorms and its properties. Fuzzy Sets Syst. 325, 35–46 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aşıcı, E.: Some notes on the \(F\)-partial order. In: Kacprzyk, J., Szmidt, E., Zadroźny, S., Atanassov, K., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology, IWIFSGN 2017, EUSFLAT 2017. Advances in Intelligent Systems and Computing, vol. 641, pp. 78–84. Springer, Cham (2018)Google Scholar
  4. 4.
    Aşıcı, E.: Some remarks on an order induced by uninorms. In: Kacprzyk, J., Szmidt, E., Zadroźny, S., Atanassov, K., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology, IWIFSGN 2017, EUSFLAT 2017. Advances in Intelligent Systems and Computing, vol. 641, pp. 69–77. Springer, Cham (2018)Google Scholar
  5. 5.
    Aşıcı, E.: An extension of the ordering based on nullnorms, Kybernetika, acceptedGoogle Scholar
  6. 6.
    Birkhoff, G.: Lattice Theory, 3rd edn, Providence (1967)Google Scholar
  7. 7.
    Calvo, T.: On some solutions of the distributivity equation. Fuzzy Sets Syst. 104, 85–96 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Calvo, T., De Baets, B., Fodor, J.: The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets Syst. 120, 385–394 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Casasnovas, J., Mayor, G.: Discrete t-norms and operations on extended multisets. Fuzzy Sets Syst. 159, 1165–1177 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    De Baets, B., Mesiar, R.: Triangular norms on the real unit square. In: Proceedings of the 1999 EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain, pp. 351–354 (1999)Google Scholar
  11. 11.
    Çaylı, G.D.: On the structure of uninorms on bounded lattices. Fuzzy Sets Syst. 357, 2–26 (2019)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Çaylı, G.D., Karaçal, F., Mesiar, R.: On internal and locally internal uninorms on bounded lattices. Int. J. Gen. Syst. 48(3), 235–259 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Çaylı, G.D.: A characterization of uninorms on bounded lattices by means of triangular norms and triangular conorms. Int. J. Gen. Syst. 47, 772–793 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Çaylı, G.D., Drygaś, P.: Some properties of idempotent uninorms on a special class of bounded lattices. Inf. Sci. 422, 352–363 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Drewniak, J., Drygaś, P., Rak, E.: Distributivity between uninorms and nullnorms. Fuzzy Sets Syst. 159, 1646–1657 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Drygaś, P.: Distributive between semi-t-operators and semi-nullnorms. Fuzzy Sets Syst. 264, 100–109 (2015)CrossRefGoogle Scholar
  17. 17.
    Drygaś, P.: A characterization of idempotent nullnorms. Fuzzy Sets Syst. 145, 455–461 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  19. 19.
    Karaçal, F., Ince, M.A., Mesiar, R.: Nullnorms on bounded lattice. Inf. Sci. 325, 227–236 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Karaçal, F., Kesicioğlu, M.N.: A T-partial order obtained from t-norms. Kybernetika 47, 300–314 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kesicioğlu, M.N.: On the property of \(T\)-distributivity. Fixed Point Theory Appl. 2013, 32 (2013)Google Scholar
  22. 22.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  23. 23.
    Mas, M., Mayor, G., Torrens, J.: T-operators. Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 7, 31–50 (1999)CrossRefGoogle Scholar
  24. 24.
    Mas, M., Mayor, G., Torrens, J.: The distributivity condition for uninorms and t-operators. Fuzzy Sets Syst. 128, 209–225 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mas, M., Mayor, G., Torrens, J.: The modularity condition for uninorms and t-operators. Fuzzy Sets Syst. 126, 207–218 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Xie, A., Liu, H.: On the distributivity of uninorms over nullnorms. Fuzzy Sets Syst. 211, 62–72 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Software Engineering, Faculty of TechnologyKaradeniz Technical UniversityTrabzonTurkey
  2. 2.Department of Mathematics and Descriptive Geometry, Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovakia

Personalised recommendations