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Uninorms on Bounded Lattices with Given Underlying Operations

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New Trends in Aggregation Theory (AGOP 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 981))

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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.

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References

  1. De Baets, B.: Idempotent uninorms. Eur. J. Oper. Res. 118, 631–642 (1999)

    Article  Google Scholar 

  2. Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publishers, Providence (1967)

    MATH  Google Scholar 

  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. 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. 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. Çayli, G.D., Drygaś, P.: Some properties of idempotent uninorms on a special class of bounded lattices. Inf. Sci. 422, 352–363 (2018)

    Article  MathSciNet  Google Scholar 

  7. Çayli, G.D., Karaçal, F.: Construction of uninorms on bounded lattices. Kybernetika 53, 394–417 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Çayli, G.D., Karaçal, F., Mesiar, R.: On a new class of uninorms on bounded lattices. Inf. Sci. 367–368, 221–231 (2016)

    Article  Google Scholar 

  9. Czogała, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)

    Article  MathSciNet  Google Scholar 

  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. Deschrijver, G.: Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory. Inf. Sci. 244, 48–59 (2013)

    Article  MathSciNet  Google Scholar 

  12. Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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. Kalina, M.: On uninorms and nullnorms on direct product of bounded lattices. Open Phys. 14(1), 321–327 (2016)

    Article  Google Scholar 

  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. Karaçal, F., Mesiar, R.: Uninorms on bounded lattices. Fuzzy Sets Syst. 261, 33–43 (2015)

    Article  MathSciNet  Google Scholar 

  18. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer, Berlin (2000)

    Book  Google Scholar 

  19. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces, North Holland, New York (1983)

    Google Scholar 

  20. Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80, 111–120 (1996)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University-Kingsville, MSC 172, Kingsville, TX, 78363, USA

    Slavka Bodjanova

  2. Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 810 05, Bratislava, Slovakia

    Martin Kalina

Authors
  1. Slavka Bodjanova
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  2. Martin Kalina
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Corresponding author

Correspondence to Martin Kalina .

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Editors and Affiliations

  1. Faculty of Science, Palacký University, Olomouc, Czech Republic

    Radomír Halaš

  2. Systems Research Institute, Polish Academy of Science, Warsaw, Poland

    Marek Gagolewski

  3. Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar

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Bodjanova, S., Kalina, M. (2019). Uninorms on Bounded Lattices with Given Underlying Operations. In: Halaš, R., Gagolewski, M., Mesiar, R. (eds) New Trends in Aggregation Theory. AGOP 2019. Advances in Intelligent Systems and Computing, vol 981. Springer, Cham. https://doi.org/10.1007/978-3-030-19494-9_17

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