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Uninorms on Bounded Lattices – Recent Development

  • Slavka Bodjanova
  • Martin KalinaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 641)

Abstract

The main goal of this paper is to explore whether on every bonded lattice L, possessing incomparable elements, one can choose incomparable elements \(\mathbf {e}\) and \(\mathbf {a}\) and then to construct a uninorm on L having \(\mathbf {e}\) and \(\mathbf {a}\) as its neutral and absorbing elements, respectively. Some necessary and some sufficient conditions for construction of uninorms on L which are neither conjunctive nor disjunctive, are given. Example of an infinite bounded lattice on which only conjunctive and disjunctive uninorms exist is presented.

Keywords

Bounded lattice Conjunctive uninorm Disjunctive uninorm Uninorm on bounded lattice Uninorm which is neither conjunctive nor disjunctive 

Notes

Acknowledgement

The work of Martin Kalina has been supported from the Science and Technology Assistance Agency under contract No. APVV-14-0013, and from the VEGA grant agency, grant No. 2/0069/16.

References

  1. 1.
    Birkhoff, G.: Lattice Theory, vol. 25, 3rd edn. American Mathematical Society Colloquium Publications, Providence (1963)zbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Calvo, T., De Baets, B., Fodor, J.: The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets Syst. 120, 385–394 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Czogała, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deschrijver, G.: A representation of t-norms in interval valued \(L\)-fuzzy set theory. Fuzzy Sets Syst. 159, 1597–1618 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deschrijver, G.: Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory. Inf. Sci. 244, 48–59 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)CrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Ince, M.A., Karaçal, F., Mesiar, R.: Some notes on nullnorms on bounded lattices. In: Kiss, G., Marichal, J.L., Teheux, B. (eds.) International Symposium on Aggregation and Structures, ISAS 2016, Book of Abstracts, p. 46 (2016)Google Scholar
  12. 12.
    Kalina, M.: On uninorms and nullnorms on direct product of bounded lattices. Open Phys. 14(1), 321–327 (2016)CrossRefGoogle Scholar
  13. 13.
    Kalina, M., Král, P.: Uninorms on Interval-Valued Fuzzy Sets. In: Carvalho, J.P., Lesot, M.-J., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R.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
  14. 14.
    Karaçal, F., Ince, M.A., Mesiar, R.: Nullnorms on bounded lattices. Inf. Sci. 325, 227–236 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Karaçal, F., Mesiar, R.: Uninorms on bounded lattices. Fuzzy Sets Syst. 261, 33–43 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  17. 17.
    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North Holland, New York (1983)zbMATHGoogle Scholar
  18. 18.
    Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80, 111–120 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

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

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