Kybernetika 53 no. 5, 877-891, 2017

About the equivalence of nullnorms on bounded lattice

M. Nesibe KesicioğluDOI: 10.14736/kyb-2017-5-0877

Abstract:

In this paper, an equivalence on the class of nullnorms on a bounded lattice based on the equality of the orders induced by nullnorms is introduced. The set of all incomparable elements w.r.t. the order induced by nullnorms is investigated. Finally, the recently posed open problems have been solved.

Keywords:

nullnorm, nounded lattice, partial order, equivalence

Classification:

03E72, 03B52

paper.pdf

References:

  1. E. Aşıcı: An order induced by nullnorms and its properties. Fuzzy Sets and Systems 325 (2017), 35-46.   DOI:10.1016/j.fss.2016.12.004
  2. M. Baczyński and B. Jayaram: Fuzzy Implications. Springer 2008.   CrossRef
  3. G. Birkhoff: Lattice Theory. Providence 1967.   DOI:10.1090/coll/025
  4. T. Calvo, B. De Baets and J. Fodor: The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets and Systems 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. Inform. Sci. 36 (1985), 85-121.   DOI:10.1016/0020-0255(85)90027-1
  6. Ü. Ertuğrul, M. N. Kesicioğlu and F. Karaçal: Ordering based on uninorms. Inform. Sci. 330 (2016), 315-327.   DOI:10.1016/j.ins.2015.10.019
  7. M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap: Aggregation Functions. Cambridge University Press 2009.   CrossRef
  8. D. Hliněná, M. Kalina and P. Král: Pre-orders and orders generated by conjunctive uninorms. In: 12th International Conference on Fuzzy Set Theory and Its Applications. Communications in Computer and Inform. Sci. 444 (2014), 307-316.   CrossRef
  9. F. Karaçal, M. A. Ince and R. Mesiar: Nullnorms on bounded lattices. Inform. Sci. 325 (2015), 227-236.   CrossRef
  10. F. Karaçal and R. Mesiar: Uninorms on bounded lattices. Fuzzy Sets and Systems 261 (2015), 33-43.   DOI:10.1016/j.fss.2014.05.001
  11. F. Karaçal and M. N. Kesicioğlu: A T-partial order obtained from t-norms. Kybernetika 47 (2011), 300-314.   CrossRef
  12. M. N. Kesicioğlu, Ü. Ertuğrul and F. Karaçal: An Equivalence relation based on the U-partial order. Inform. Sci. 411 (2017), 39-51.   DOI:10.1016/j.ins.2017.05.020
  13. M. N. Kesicioğlu, F. Karaçal and R. Mesiar: Order-equivalent triangular norms. Fuzzy Sets and Systems 268 (2015), 59-71.   CrossRef
  14. M. N. Kesicioğlu and R. Mesiar: Ordering based on implications. Inform. Sci. 276 (2014), 377-386.   CrossRef
  15. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers 2000.   DOI:10.1007/978-94-015-9540-7
  16. Z. Ma and W. M. Wu: Logical operators on complete lattices. Inform. Sci. 55 (1991), 77-97.   CrossRef
  17. M. Mas, G. Mayor and J. Torrens: t-operators. Int. J. Uncertainty Fuzziness Knowledge-Based Syst. 7 (1999), 31-50.   DOI:10.1142/s0218488599000039
  18. M. Mas, G. Mayor and J. Torrens: The modularity condition for uninorms nd t-operators. Fuzzy Sets and Systems 126 (2002), 207-218.   DOI:10.1016/s0165-0114(01)00055-0
  19. R. R. Yager: Aggregation operators and fuzzy systems modelling. Fuzzy Sets and Systems 67 (1994), 129-145.   DOI:10.1016/0165-0114(94)90082-5