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On the F-partial Order and Equivalence Classes of Nullnorms

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Aggregation Functions in Theory and in Practice (AGOP 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 581))

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Abstract

In this paper, we define the set \(I_{F}^{(x)}\), denoting the set of all incomparable elements with arbitrary but fixed \(x\in (0,1)\) according to F-partial order and this set is deeply investigated. Then, an equivalence relation on the class of nullnorms induced by a F-partial order is defined and discussed. Finally, we give an answer to a recently posed open problem.

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References

  1. Aşıcı, E., Karaçal, F.: On the \(T\)-partial order and properties. Inf. Sci. 267, 323–333 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aşıcı, E., Karaçal, F.: Incomparability with respect to the triangular order. Kybernetika 52, 15–27 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Aşıcı, E.: An order induced by nullnorms and its properties. Fuzzy Sets Syst. doi:10.1016/j.fss.2016.12.004

  4. Aşıcı, E.: The equivalence classes of nullnorms. Working paper

    Google Scholar 

  5. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casasnovas, J., Mayor, G.: Discrete t-norms and operations on extended multisets. Fuzzy Sets Syst. 159, 1165–1177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Baets, B., Mesiar, R.: Triangular norms on the real unit square. In: Proceedings of the 1999 EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain, pp. 351–354 (1999)

    Google Scholar 

  9. Çaylı, 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 

  10. Drewniak, J., Drygaś, P., Rak, E.: Distributivity between uninorms and nullnorms. Fuzzy Sets Syst. 159, 1646–1657 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Drygaś, P.: A characterization of idempotent nullnorms. Fuzzy Sets Syst. 145, 455–461 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karaçal, F., Ince, M.A., Mesiar, R.: Nullnorms on bounded lattice. Inf. Sci. 325, 227–236 (2015)

    Article  MathSciNet  Google Scholar 

  13. Karaçal, F., Kesicioğlu, M.N.: A T-partial order obtained from t-norms. Kybernetika 47, 300–314 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Kesicioğlu, M.N., Karaçal, F., Mesiar, R.: Order-equivalent triangular norms. Fuzzy Sets Syst. 268, 59–71 (2015)

    Article  MathSciNet  Google Scholar 

  15. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  16. Liang, X., Pedrycz, W.: Logic-based fuzzy networks: a study in system modeling with triangular norms and uninorms. Fuzzy Sets Syst. 160, 3475–3502 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mas, M., Mayor, G., Torrens, J.: t-Operators. Int. J. Uncertain. Fuzz Knowl.-Based Syst. 7, 31–50 (1999)

    Article  MATH  Google Scholar 

  18. Mas, M., Mayor, G., Torrens, J.: The distributivity condition for uninorms and t-operators. Fuzzy Sets Syst. 128, 209–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mitsch, H.: A natural partial order for semigroups. Proc. Am. Math. Soc. 97, 384–388 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Saminger, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets Syst. 157, 1403–1416 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xie, A., Liu, H.: On the distributivity of uninorms over nullnorms. Fuzzy Sets Syst. 211, 62–72 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

In this paper, the full proofs are contained in [4].

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Authors and Affiliations

  1. Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, 61830, Trabzon, Turkey

    Emel Aşıcı

Authors
  1. Emel Aşıcı
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Correspondence to Emel Aşıcı .

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Editors and Affiliations

  1. School of Informatics, University of Skövde, Skövde, Sweden

    Vicenç Torra

  2. Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar

  3. KERMIT Research Unit Knowledge-based Systems, Ghent University, Ghent, Belgium

    Bernard De Baets

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Aşıcı, E. (2018). On the F-partial Order and Equivalence Classes of Nullnorms. In: Torra, V., Mesiar, R., Baets, B. (eds) Aggregation Functions in Theory and in Practice. AGOP 2017. Advances in Intelligent Systems and Computing, vol 581. Springer, Cham. https://doi.org/10.1007/978-3-319-59306-7_16

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  • DOI: https://doi.org/10.1007/978-3-319-59306-7_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59305-0

  • Online ISBN: 978-3-319-59306-7

  • eBook Packages: EngineeringEngineering (R0)

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