Kybernetika 60 no. 1, 76-89, 2024

Representation of uni-nullnorms and null-uninorms on bounded lattices

Yi-Qun Zhang, Ya-Ming Wang and Hua-Wen LiuDOI: 10.14736/kyb-2024-1-0076

Abstract:

In this paper, we present the representation for uni-nullnorms with disjunctive underlying uninorms on bounded lattices. It is shown that our method can cover the representation of nullnorms on bounded lattices and some of existing construction methods for uni-nullnorms on bounded lattices. Illustrative examples are presented simultaneously. In addition, the representation of null-uninorms with conjunctive underlying uninorms on bounded lattices is obtained dually.

Keywords:

uninorm, bounded lattice, T-norm, uni-nullnorm, null-uninorm

Classification:

03E72, 03B52, 03G10, 18B35

paper.pdf

References:

  1. P. Akella: Structure of n-uninorms. Fuzzy Sets Syst. 158 (2007), 1631-1651.   DOI:10.1016/j.fss.2007.02.015
  2. B. C. Bedregal, H. S. Santos and R. Callejas-Bedregal: T-norms on bounded lattices: t-norm morphisms and operators. In: IEEE International Conference on Fuzzy Systems, 2006.   CrossRef
  3. S. Bodjanova and M. Kalina: Construction of uninorms on bounded lattices. In: IEEE 12th International Symposium on Intelligent Systems and Informatics, SISY 2014, Subotica 2014.   CrossRef
  4. T. Calvo, B. De Baets and J. Fodor: The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets Syst. 120 (2001), 385-394.   DOI:10.1016/S0165-0114(99)00125-6
  5. D. Dubois and H. Prade: A review of fuzzy set aggregation connectives. Inf. Sci. 36 (1985), 85-121.   DOI:10.1016/0020-0255(85)90027-1
  6. Ü. Ertuğrul, M. N. Kesicioğlu and F. Karaçal: Construction methods for uni-nullnorms and null-uninorms on bounded lattice. Kybernetika 55 (2019), 994-1015.   DOI:10.14736/kyb-2019-6-0994
  7. M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap: Aggregation Functions. Cambridge University Press, Cambridge 2009.   CrossRef
  8. F. Karaçal, M. A. Ince and R. Mesiar: Nullnorms on bounded lattices. Inf. Sci. 325 (2015), 227-236.   DOI:10.1016/j.ins.2015.06.052
  9. F. Karaçal and R. Mesiar: Uninorms on bounded lattices. Fuzzy Sets Syst. 261 (2015), 33-43.   DOI:10.1016/j.fss.2014.05.001
  10. K. Menger: Statistical metrics. Proc. Natl. Acad. Sci. USA 28 (1942), 535-537.   DOI:10.1073/pnas.28.12.535
  11. F. Sun, X. B. Qu and X. J. Huang: Characterizations of residual implications derived from uni-nullnorms. Comput. Appl. Math. 39 (2020), 1-18.   DOI:10.1007/s40314-020-1135-7
  12. F. Sun, X. P. Wang and X. B. Qu: Uni-nullnorms and null-uninorms. J. Intell. Fuzzy Syst. 32 (2017), 1969-1981.   DOI:10.3233/jifs-161495
  13. F. Sun, X. P. Wang and X. B. Qu: Characterizations of uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms. Fuzzy Sets Syst. 334 (2018), 24-35.   DOI:10.1016/j.fss.2017.03.001
  14. X. R. Sun and H. W. Liu: Representation of nullnorms on bounded lattices. Inf. Sci. 539 (2020), 269-276.   DOI:10.1016/j.ins.2020.06.013
  15. G. Wang, F. Qin and W. H. Li: Distributivity and conditional distributivity for uni-nullnorms. Fuzzy Sets Syst. 372 (2019), 34-49.   DOI:10.1016/j.fss.2018.08.008
  16. Y. M. Wang, H. Zhan and H. W. Liu: Uni-nullnorms on bounded lattices. Fuzzy Sets Syst. 386 (2020), 132-144.   DOI:10.1016/j.fss.2019.01.001
  17. Y. M. Wang, Y. Y. Zhao and H. W. Liu: Methods for obtaining uni-nullnorms and null-uninorms on bounded lattices. Fuzzy Sets Syst. 441 (2022), 279-285.   DOI:10.1016/j.fss.2022.02.017
  18. X. Wu, S. Liang, Y. Luo and G. D. Çaylı: Construction methods for the smallest and largest uni-nullnorms on bounded lattices. Fuzzy Sets Syst. 427 (2022), 132-148.   DOI:10.1016/j.fss.2021.02.010
  19. R. R. Yager: Aggregation operators and fuzzy systems modelling. Fuzzy Sets Syst. 67 (1994), 129-145.   DOI:10.1016/0165-0114(94)90082-5
  20. R. R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets Syst. 80 (1996), 111-120.   DOI:10.1016/0165-0114(95)00133-6
  21. R. R. Yager: Defending against strategic manipulation in uninorm-based multi-agent decision making. Eur. J. Oper. Res. 141 (2002), 217-232.   DOI:10.1016/S0377-2217(01)00267-3
  22. H. P. Zhang, Y. Ouyang and B. De Baets: Constructions of uni-nullnorms and null-uninorms on a bounded lattice. Fuzzy Sets Syst. 403 (2021), 78-87.   DOI:10.1016/j.fss.2019.10.010
  23. H. P. Zhang, Y. Ouyang, Z. Wang and B. De Baets: A complete representation theorem for nullnorms on bounded lattices with ample illustrations. Fuzzy Sets Syst. 439 (2022), 157-169.   DOI:10.1016/j.fss.2021.08.012
  24. T. H. Zhang, F. Qin and W. H. Li: On the distributivity equations between uni-nullnorms and overlap (grouping) functions. Fuzzy Sets Syst. 403 (2021), 56-77.   DOI:10.1016/j.fss.2019.12.005