Kybernetika 55 no. 3, 518-530, 2019

Some notes on U-partial order

M. Nesibe Kesicioğlu, Ü. Ertuğrul and F. KaraçalDOI: 10.14736/kyb-2019-3-0518


In this paper, an equivalence on the class of uninorms on a bounded lattice is discussed. Some relationships between the equivalence classes of uninorms and the equivalence classes of their underlying t-norms and t-conorms are presented. Also, a characterization for the sets admitting some incomparability w.r.t. the U-partial order is given.


uninorm, bounded lattice, partial order, equivalence, T-norm


03E72, 03B52


  1. E. Aşıcı and F. Karaçal: On the T-partial order and properties. Inform. Sci. 267 (2014), 323-333.   DOI:10.1016/j.ins.2014.01.032
  2. M. Baczyński and B. Jayaram: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231, Springer, Berlin, Heidelberg, 2008.   CrossRef
  3. G. Birkhoff: Lattice Theory. Third edition. Providence, 1967.   DOI:10.1090/coll/025
  4. T. Calvo, G. Mayor and R. Mesiar: Aggregation operators. New Trends and Applications. Studies in Fuzziness and Soft Computing, Physica-Verlag HD, Heidelberg, 2002.   DOI:10.1007/978-3-7908-1787-4
  5. P. Drygaś, D. Ruiz-Aguilera and J. Torrens: A characterization of a class of uninorms with continuous underlying operators. Fuzzy Sets and Systems 287 (2016), 137-153.   DOI:10.1016/j.fss.2015.07.015
  6. U. 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. J. Fodor, R. Yager and A. Rybalov: Structure of uninorm. Int. J. Uncertain. Fuzziness Knowledge-Based Systems 5 (1997), 411-427.   DOI:10.1142/s0218488597000312
  8. M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap: Aggregation Functions. Cambridge University Press, 2009.   CrossRef
  9. D. Hliněná, M. Kalina and P. Král: Pre-orders and orders generated by conjunctive uninorms. In: Inf. Proc. Manage. of Uncert. Knowledge-Based Syst. Communications in Computer and Inf. Sci. 444 (2014), pp. 307-316.   CrossRef
  10. F. Karaçal and R. Mesiar: Uninorms on bounded lattices. Fuzzy Sets and Systems 261 (2015), 33-43.   CrossRef
  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: Some notes on the partial orders induced by a uninorm and a nullnorm in a bounded lattice. Fuzzy Sets and Systems 346 (2018), 59-71.   DOI:10.1016/j.fss.2014.10.006
  13. 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
  14. M. N. Kesicioğlu, F. Karaçal and R. Mesiar: Order-equivalent triangular norms. Fuzzy Sets and Systems 268 (2015), 59-71.   DOI:10.1016/j.fss.2014.10.006
  15. M. N. Kesicioğlu and R. Mesiar: Ordering based on implications. Inform. Sci. 276 (2014), 377-386.   DOI:10.1016/j.ins.2013.12.047
  16. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  17. J. Lu, K. Wang and B. Zhao: Equivalence relations induced by the U-partial order. Fuzzy Sets and Systems 334 (2018), 73-82.   DOI:10.1016/j.fss.2017.07.013
  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. M. Mas, S. Massanet, D. Ruiz-Aguilera and J. Torrens: A survey on the existing classes of uninorms. J. Intell. Fuzzy Syst. 29 (2015), 1021-1037.   DOI:10.3233/ifs-151728
  20. S. Saminger: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems 157 (2006), 1403-1416.   DOI:10.1016/j.fss.2005.12.021
  21. R. R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111-120.   DOI:10.1016/0165-0114(95)00133-6
  22. 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
  23. R. R. Yager: Uninorms in fuzzy system modelling. Fuzzy Sets and Systems 122 (2001), 167-175.   DOI:10.1016/s0165-0114(00)00027-0