Kybernetika 53 no. 5, 911-921, 2017

Notes on locally internal uninorm on bounded lattices

Gül Deniz Çaylı, Ümit Ertuğrul, Tuncay Köroğlu and Funda KaraçalDOI: 10.14736/kyb-2017-5-0911


In the study, we introduce the definition of a locally internal uninorm on an arbitrary bounded lattice $L$. We examine some properties of an idempotent and locally internal uninorm on an arbitrary bounded latice $L$, and investigate relationship between these operators. Moreover, some illustrative examples are added to show the connection between idempotent and locally internal uninorm.


uninorm, idempotent uninorm, bounded lattice, locally internal


03B52, 06B20, 03E72


  1. E. Aşıcı and F. Karaçal: Incomparability with respect to the triangular order. Kybernetika 52 (2016), 15-27.   DOI:10.14736/kyb-2016-1-0015
  2. G. Birkhoff: Lattice Theory. American Mathematical Society Colloquium Publ., Providence 1967.   DOI:10.1090/coll/025
  3. B. De Baets: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631-642.   DOI:10.1016/s0377-2217(98)00325-7
  4. B. De Baets and J. Fodor: A single-point characterization of representable uninorms. Fuzzy Sets Syst. 202 (2012), 89-99.   DOI:10.1016/j.fss.2011.12.001
  5. G. D. Çaylı, F. Karaçal and R. Mesiar: On a new class of uninorms on bounded lattices. Inform. Sci. 367-368 (2016), 221-231.   DOI:10.1016/j.ins.2016.05.036
  6. G. D. Çaylı and P. Drygaś: Some properties of idempotent uninorms on bounded lattices. Inform. Sci. 422 (2018), 352-363.   DOI:10.1016/j.ins.2017.09.018
  7. J. Drewniak and P. Drygaś: On a class of uninorms. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 10 (2002), 5-10.   DOI:10.1142/s021848850200179x
  8. P. Drygaś: On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums. Fuzzy Sets Syst. 161 (2010), 149-157.   DOI:10.1016/j.fss.2009.09.017
  9. P. Drygaś, D. Ruiz-Aguilera and J. Torrens: A characterization of uninorms locally internal in $A(e)$ with continuous underlying operators. Fuzzy Sets Syst. 287 (2016), 137-153.   DOI:10.1016/j.fss.2009.09.017
  10. 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
  11. J. Fodor, R. R. Yager and A. Rybalov: Structure of uninorms. Int. J. Uncertain Fuzziness Knowl.-Based Syst. 5 (1997), 411-427.   DOI:10.1142/s0218488597000312
  12. M. N. Kesicioğlu and R. Mesiar: Ordering based on implications. Fuzzy Sets Syst. 276 (2014), 377-386.   DOI:10.1016/j.ins.2013.12.047
  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. F. Karaçal and R. Mesiar: Uninorms on bounded lattices. Fuzzy Sets Syst. 261 (2015), 33-43.   DOI:10.1016/j.fss.2014.05.001
  15. F. Karaçal, Ü. Ertuğrul and R. Mesiar: Characterization of uninorms on bounded lattices. Fuzzy Sets Syst. 308 (2017), 54-71.   DOI:10.1016/j.fss.2016.05.014
  16. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Dordrecht, Kluwer Acad. Publ., 2000.   DOI:10.1007/978-94-015-9540-7
  17. J. Martin, G. Mayor and J. Torrens: On locally internal monotonic operations. Fuzzy Sets Syst. 137 (2003), 27-42.   DOI:10.1016/s0165-0114(02)00430-x
  18. A. Mesiarová-Zemánková: Multi-polar t-conorms and uninorms. Inform. Sci. 301 (2015), 227-240.   DOI:10.1016/j.ins.2014.12.060
  19. R. R. Yager: Uninorms in fuzzy system modeling. Fuzzy Sets Syst. 122 (2001), 167-175.   DOI:10.1016/s0165-0114(00)00027-0
  20. R. R. Yager: Defending against strategic manipulation in uninorm-based multi-agent decision making. European J. Oper. Res. 141 (2002), 217-232.   DOI:10.1016/s0377-2217(01)00267-3
  21. R. R. Yager and A. Rybalov: Uninorms aggregation operators. Fuzzy Sets Syst. 80 (1996), 111-120.   DOI:10.1016/0165-0114(95)00133-6