Advertisement

Uninorms on Bounded Lattices with Given Underlying Operations

  • Slavka Bodjanova
  • Martin KalinaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 981)

Abstract

The main goal of this paper is to construct uninorms on a bounded lattice L which are neither conjunctive nor disjunctive (i.e., are of the third type), and have a given underlying t-norm and t-conorm. Three different cases will be studied. The first one will present some (quite general) conditions under which it is possible to construct a uninorm of the third type, regardless of the type of underlying t-norm and t-conorm. Then, two different cases of idempotent uninorm will be presented. Finally, a uninorm with a given underlying t-norm and t-conorm will be presented.

Keywords

Bounded lattice Idempotent uninorm t-conorm t-norm Uninorm Uninorm of the third type Uninorm which is neither conjunctive nor disjunctive 

Notes

Acknowledgements

The work of Martin Kalina has been supported from the VEGA grant agency, grant No. 2/0069/16 and 1/0006/19.

References

  1. 1.
    De Baets, B.: Idempotent uninorms. Eur. J. Oper. Res. 118, 631–642 (1999)CrossRefGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publishers, Providence (1967)zbMATHGoogle Scholar
  3. 3.
    Bodjanova, S., Kalina, M.: Construction of uninorms on bounded lattices. In: IEEE 12th International Symposium on Intelligent Systems and Informatics, SISY 2014, Subotica, Serbia, pp. 61–66 (2014)Google Scholar
  4. 4.
    Bodjanova, S., Kalina, M.: Uninorms on bounded lattices - recent development. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology 2017, EUSFLAT 2017, IWIFSGN : Advances in Intelligent Systems and Computing, vol. 641. Springer, Cham (2017)Google Scholar
  5. 5.
    Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R.: Aggregation operators: properties, classes and construction methods. In: Aggregation Operators, pp. 3–104. Physica-Verlag GMBH, Heidelberg (2002)Google Scholar
  6. 6.
    Çayli, G.D., Drygaś, P.: Some properties of idempotent uninorms on a special class of bounded lattices. Inf. Sci. 422, 352–363 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Çayli, G.D., Karaçal, F.: Construction of uninorms on bounded lattices. Kybernetika 53, 394–417 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Çayli, G.D., Karaçal, F., Mesiar, R.: On a new class of uninorms on bounded lattices. Inf. Sci. 367–368, 221–231 (2016)CrossRefGoogle Scholar
  9. 9.
    Czogała, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Deschrijver, G.: A representation of t-norms in interval valued \(L\)-fuzzy set theory. Fuzzy Sets Syst. 159, 1597–1618 (2008)Google Scholar
  11. 11.
    Deschrijver, G.: Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory. Inf. Sci. 244, 48–59 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dombi, J.: A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 8, 149–163 (1982)CrossRefGoogle Scholar
  14. 14.
    Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation functions. In: Encyclopedia of Mathematics and Its Applications, vol. 127. Cambridge University Press, Cambridge (2009)Google Scholar
  15. 15.
    Kalina, M.: On uninorms and nullnorms on direct product of bounded lattices. Open Phys. 14(1), 321–327 (2016)CrossRefGoogle Scholar
  16. 16.
    Kalina, M., Král’, P.: Uninorms on interval-valued fuzzy sets. In: Carvalho, J., Lesot, M.J., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2016, Communications in Computer and Information Science, vol. 611, pp. 522–531. Springer, Cham (2016)Google Scholar
  17. 17.
    Karaçal, F., Mesiar, R.: Uninorms on bounded lattices. Fuzzy Sets Syst. 261, 33–43 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer, Berlin (2000)CrossRefGoogle Scholar
  19. 19.
    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces, North Holland, New York (1983)Google Scholar
  20. 20.
    Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80, 111–120 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M University-KingsvilleKingsvilleUSA
  2. 2.Department of Mathematics, Faculty of Civil EngineeringSlovak University of Technology in BratislavaBratislavaSlovakia

Personalised recommendations