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Aggregation Functions on Bounded Posets

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Book cover 35 Years of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 261))

Abstract

A general concept of aggregation on bounded posets is introduced and discussed. Up to general results, several particular results due to E.E. Kerre and his research group are recalled in the light of our approach. A special attention is paid to triangular norms on bounded posets.

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References

  1. Alsina, C., Frank, M.J., Schweizer, B.: Associative functions. World Scientific Publishing Co. Pte. Ltd., Hackensack (2006)

    Book  MATH  Google Scholar 

  2. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, 87–96 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Birkhoff, G.: Lattice theory, 3rd edn. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  4. Bustince, H., Barrenechea, E., Pagola, M.: Generation of interval-valued fuzzy and Atanassov’s intuitionistic fuzzy connectives from fuzzy connectives and from K– operators: Laws for conjunctions and disjunctions, amplitude. International Journal of Intelligent Systems 23(6), 680–714 (2008)

    Article  MATH  Google Scholar 

  5. Bustince, H., Montero, J., Pagola, M., Barrenechea, E., Gómez, D.: A survey on interval-valued fuzzy sets. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Handbook of granular computing, pp. 489–515. Wiley, New York (2008)

    Chapter  Google Scholar 

  6. Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R.: Aggregation operators: Properties, classes and construction methods. In: Calvo, T., Mayor, G., Mesiar, R. (eds.) Aggregation operators: Properties, Classes and Construction Methods, Aggregation Operators: New Trends and Applications, pp. 1–104. Physica-Verlag, Heilderberg (2002)

    Google Scholar 

  7. Clifford, A.H.: Naturally totally ordered commutative semigroups. Amer. J. Math. 76, 631–646 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  8. De Baets, B., Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104, 61–76 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Cooman, G., Kerre, E.E.: Order Norms On Bounded Partially Ordered Sets. The Journal of Fuzzy Mathematics 2, 281–310 (1994)

    MATH  Google Scholar 

  10. Demirci, M.: Aggregation operators on partially ordered sets and their categorical foundations. Kybernetika 42, 261–277 (2006)

    MathSciNet  Google Scholar 

  11. Deschrijver, G.: Additive and multiplicative generators in interval-valued fuzzy set theory. IEEE Transactions on Fuzzy Systems 15(2), 222–237 (2007)

    Article  Google Scholar 

  12. Deschrijver, G.: Arithmetic operators in interval-valued fuzzy set theory. Information Sciences 177(14), 2906–2924 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Deschrijver, G.: Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory. Fuzzy Sets and Systems 160(6), 778–801 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Deschrijver, G.: Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory. Fuzzy Sets and Systems 160(21), 3080–3102 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Deschrijver, G., Cornelis, C., Kerre, E.E.: Intuitionistic fuzzy connectives revisited. In: Proc. 9th Internat. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Annecy, France, pp. 1839–1844 (2002)

    Google Scholar 

  16. Deschrijver, G., Cornelis, C., Kerre, E.E.: On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems 12(1), 45–61 (2004)

    Article  MathSciNet  Google Scholar 

  17. Deschrijver, G., Kerre, E.E.: Classes of intuitionistic fuzzy t-norms satisfying the residuation principle. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11(6), 691–709 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Deschrijver, G., Kerre, E.E.: On the Relationship Between some Extensions of Fuzzy Set Theory. Fuzzy Sets and Systems 133, 227–235 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Deschrijver, G., Kerre, E.E.: Uninorms in L *–fuzzy set theory. Fuzzy Sets and Systems 148, 243–262 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Deschrijver, G., Kerre, E.E.: Triangular norms and related operators in L ∗ -fuzzy set theory. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, pp. 231–259. Elsevier, Amsterdam (2005)

    Chapter  Google Scholar 

  21. Deschrijver, G., Kerre, E.: Aggregation operators in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory. Studies in Fuzziness and Soft Computing 220, 183–203 (2008)

    Article  Google Scholar 

  22. Dimuro, G.P., Bedregal, B.C., Reiser, R.H.S., Santiago, R.H.N.: Interval additive generators of interval t-norms. In: Hodges, W., de Queiroz, R. (eds.) Logic, Language, Information and Computation. LNCS (LNAI), vol. 5110, pp. 123–135. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  23. Dubois, D., Prade, H.: Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory. In: Fuzzy sets and decision analysis 20 of Stud. Management Sci., pp. 209–240. North-Holland, Amsterdam (1984)

    Google Scholar 

  24. Durante, F., Mesiar, R., Papini, P.-L.: The lattice-theoretic structure of the sets of triangular norms and semi-copulas. Nonlinear Anal. 69(1), 46–52 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Durante, F., Sempi, C.: Semicopula. Kybernetika 41(3), 315–328 (2005)

    MathSciNet  Google Scholar 

  26. Fung, L.W., Fu, K.S.: An axiomatic approach to rational decision making in a fuzzy environment. In: Fuzzy sets and their applications to cognitive and decision processes (Proc. U. S.-Japan Sem., Univ. Calif., Berkeley, Calif., 1974), pp. 227–256. Academic Press, New York (1975)

    Google Scholar 

  27. Grätzer, G.: General lattice theory, 2nd edn. Birkhäuser Verlag, Basel (1998)

    MATH  Google Scholar 

  28. Grabisch, M., Marichal, J.–L., Mesiar, R., Pap, E.: Aggregation functions. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  29. Höhle, U.: Commutative, residuated l-monoids. In: Höhle, U., Klement, E.P. (eds.) Nonclassical Logics and their Applications to Fuzzy Subsets, pp. 53–106. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  30. Jenei, S., De Baets, B.: On the direct decomposability of t-norms on product lattices. Fuzzy Sets and Systems 139(3), 699–707 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Karacal, F., Khadjiev, D.: \(\bigvee\)–distributive and infinitely \(\bigvee\)–distributive t-norms on complete lattices. Fuzzy Sets and Systems 151(2), 341–352 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Trends in Logic, Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  33. Klir, G., Folger, T.: Fuzzy Sets, Uncertainty and Information. Prentice-Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  34. Kolesárová, A., Komorníková, M.: Triangular norm-based iterative compensatory operators. Fuzzy Sets and Systems 104, 109–119 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  35. Marichal, J.L.: k–intolerant capacities and Choquet integrals. European J. Oper. Res. 177(3), 1453–1468 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Saminger-Platz, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems 157(10), 1403–1416 (2006)

    Article  MathSciNet  Google Scholar 

  37. Saminger-Platz, S.: The dominance relation in some families of continuous Archimedean t–norms and copulas. Fuzzy Sets and Systems 160(14), 2017–2031 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Saminger-Platz, S., Klement, E.P., Mesiar, R.: On extensions of triangular norms on bounded lattices. Indagationes Mathematicae 19(1), 135–150 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  39. Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–330 (1960)

    MATH  MathSciNet  Google Scholar 

  40. Sugeno, M.: Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology (1974)

    Google Scholar 

  41. Turksen, I.B.: Interval-valued fuzzy sets and ’compensatory AND’. Fuzzy Sets and Systems 51(3), 295–307 (1992)

    Article  MathSciNet  Google Scholar 

  42. Zhang, D.: Triangular norms on partially ordered sets. Fuzzy Sets and Systems 153(2), 195–209 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Mesiar, R., Komorníková, M. (2010). Aggregation Functions on Bounded Posets. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds) 35 Years of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16629-7_1

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  • DOI: https://doi.org/10.1007/978-3-642-16629-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16628-0

  • Online ISBN: 978-3-642-16629-7

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