Kybernetika 57 no. 6, 989-1004, 2021

On the direct product of uninorms on bounded lattices

Emel Aşıcı and Radko MesiarDOI: 10.14736/kyb-2021-6-0989


In this paper, we study on the direct product of uninorms on bounded lattices. Also, we define an order induced by uninorms which are a direct product of two uninorms on bounded lattices and properties of introduced order are deeply investigated. Moreover, we obtain some results concerning orders induced by uninorms acting on the unit interval $ [0,1] $.


uninorm, partial order, direct product


03E72, 03B52


  1. E. Aşıcı: An order induced by nullnorms and its properties. Fuzzy Sets Syst. 325 (2017), 35-46.   DOI:10.1016/j.fss.2016.12.004
  2. E. Aşıcı: The equivalence of uninorms induced by the $ U $-partial order. Hacet. J. Math. Stat. 48 (2019), 2, 439-450.   DOI:10.15672/hjms.2019.662
  3. E. Aşıcı: Construction methods for triangular norms and triangular conorms on appropriate bounded lattices. Iran. J. Fuzzy Syst. 18 (2021), 81-98.   DOI:10.22111/IJFS.2021.6083
  4. 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
  5. E. Aşıcı and R. Mesiar: Alternative approaches to obtain t-norms and t-conorms on bounded lattices. Iran. J. Fuzzy Syst. 17 (2020), 121-138.   DOI:10.22111/IJFS.2020.5410
  6. E. Aşıcı and R. Mesiar: On generating uninorms on some special classes of bounded lattices Fuzzy Sets Syst.   CrossRef
  7. E. Aşıcı and R. Mesiar: On the construction of uninorms on bounded lattices. Fuzzy Sets Syst. 408 (2021), 65-85.   DOI:10.1016/j.fss.2020.02.007
  8. G. Beliakov, A. Pradera and T. Calvo: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221, Springer, Berlin, Heidelberg 2007.   CrossRef
  9. G. Birkhoff: Lattice Theory. Third edition. Providence 1967.   CrossRef
  10. 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
  11. T. Calvo, G. Mayor and R. Mesiar: Aggregation Operators: New Trends and Applications. Studies in Fuzziness and Soft Computing, vol. 97, Springer, Berlin, Heidelberg 2002.   CrossRef
  12. J. Casasnovas and G. Mayor: Discrete t-norms and operations on extended multisets. Fuzzy Sets Syst. 159 (2008), 1165-1177.   DOI:10.1016/j.fss.2007.12.005
  13. G. D. Çaylı: New methods to construct uninorms on bounded lattices. Int. J. Approximate Reason. 115 (2019), 254-264.   DOI:10.1016/j.ijar.2019.10.006
  14. G. D. Çaylı: Uninorms on bounded lattices with the underlying t-norms and t-conorms. Fuzzy Sets and Systems 395 (2020), 107-129.   DOI:10.1016/j.fss.2019.06.005
  15. G. D. Çaylı: Alternative approaches for generating uninorms on bounded lattices. Inform. Sci. 488 (2019), 111-139.   DOI:10.1016/j.ins.2019.03.007
  16. B. De Baets and R. Mesiar: Triangular norms on product lattices. Fuzzy Sets Syst. 104 (1999), 61-75.   DOI:10.1016/S0165-0114(98)00259-0
  17. J. Drewniak, P. Drygaś and E. Rak: Distributivity between uninorms and nullnorms. Fuzzy Sets Syst. 159 (2008), 1646-1657.   DOI:10.1016/j.fss.2007.09.015
  18. 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
  19. J. C. Fodor, R. R. Yager and A. Rybalov: Structure of uninorms. Int. J. Uncertain Fuzz. Knowl.-Based Syst. 5 (1997), 411-427.   CrossRef
  20. J. Fodor and B. De Baets: A single-point characterization of representable uninorms. Fuzzy Sets Syst. 202 (2012), 89-99.   DOI:10.1016/j.fss.2011.12.001
  21. M. Kalina: On uninorms and nullnorms on direct product of bounded lattices. Open Phys. 14 (2016), 321-327.   DOI:10.1515/phys-2016-0035
  22. F. Karaçal and R. Mesiar: Uninorms on bounded lattices. Fuzzy Sets Syst. 261 (2015), 33-43.   DOI:10.1016/j.fss.2014.05.001
  23. F. Karaçal and M. N. Kesicioğlu: A T-partial order obtained from t-norms. Kybernetika 47 (2011), 300-314.   DOI:10.1159/000337863
  24. M. N. Kesicioğlu, F. Karaçal and Ü. Ertuğrul: An equivalence relation based on the $ U $-partial order. Inform. Sci. 411 (2017), 39-51.   DOI:10.1016/j.ins.2017.05.020
  25. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  26. G. J. Klir and B. Yuan: Fuzzy Sets and Fuzzy Logic, Theory and Application. Prentice Hall PTR, Upper Saddle River, New Jersey 1995.   CrossRef
  27. S. Saminger: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets Syst. 157 (2006), 1403-1416.   DOI:10.1016/j.fss.2005.12.021
  28. R. R. Yager and A. Rybalov: Uninorm aggregation operators. Fuzzy Sets Syst. 80 (1996), 111-120.   DOI:10.1016/0165-0114(95)00133-6