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
Alsina, C., Frank, M.J., Schweizer, B.: Associative functions. World Scientific Publishing Co. Pte. Ltd., Hackensack (2006)
Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, 87–96 (1986)
Birkhoff, G.: Lattice theory, 3rd edn. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1967)
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)
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)
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)
Clifford, A.H.: Naturally totally ordered commutative semigroups. Amer. J. Math. 76, 631–646 (1954)
De Baets, B., Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104, 61–76 (1999)
De Cooman, G., Kerre, E.E.: Order Norms On Bounded Partially Ordered Sets. The Journal of Fuzzy Mathematics 2, 281–310 (1994)
Demirci, M.: Aggregation operators on partially ordered sets and their categorical foundations. Kybernetika 42, 261–277 (2006)
Deschrijver, G.: Additive and multiplicative generators in interval-valued fuzzy set theory. IEEE Transactions on Fuzzy Systems 15(2), 222–237 (2007)
Deschrijver, G.: Arithmetic operators in interval-valued fuzzy set theory. Information Sciences 177(14), 2906–2924 (2007)
Deschrijver, G.: Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory. Fuzzy Sets and Systems 160(6), 778–801 (2009)
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)
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)
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)
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)
Deschrijver, G., Kerre, E.E.: On the Relationship Between some Extensions of Fuzzy Set Theory. Fuzzy Sets and Systems 133, 227–235 (2003)
Deschrijver, G., Kerre, E.E.: Uninorms in L *–fuzzy set theory. Fuzzy Sets and Systems 148, 243–262 (2004)
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)
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)
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)
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)
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)
Durante, F., Sempi, C.: Semicopula. Kybernetika 41(3), 315–328 (2005)
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)
Grätzer, G.: General lattice theory, 2nd edn. Birkhäuser Verlag, Basel (1998)
Grabisch, M., Marichal, J.–L., Mesiar, R., Pap, E.: Aggregation functions. Cambridge University Press, Cambridge (2009)
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)
Jenei, S., De Baets, B.: On the direct decomposability of t-norms on product lattices. Fuzzy Sets and Systems 139(3), 699–707 (2003)
Karacal, F., Khadjiev, D.: \(\bigvee\)–distributive and infinitely \(\bigvee\)–distributive t-norms on complete lattices. Fuzzy Sets and Systems 151(2), 341–352 (2005)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Trends in Logic, Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)
Klir, G., Folger, T.: Fuzzy Sets, Uncertainty and Information. Prentice-Hall, Englewood Cliffs (1988)
Kolesárová, A., Komorníková, M.: Triangular norm-based iterative compensatory operators. Fuzzy Sets and Systems 104, 109–119 (1999)
Marichal, J.L.: k–intolerant capacities and Choquet integrals. European J. Oper. Res. 177(3), 1453–1468 (2007)
Saminger-Platz, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems 157(10), 1403–1416 (2006)
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)
Saminger-Platz, S., Klement, E.P., Mesiar, R.: On extensions of triangular norms on bounded lattices. Indagationes Mathematicae 19(1), 135–150 (2008)
Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–330 (1960)
Sugeno, M.: Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology (1974)
Turksen, I.B.: Interval-valued fuzzy sets and ’compensatory AND’. Fuzzy Sets and Systems 51(3), 295–307 (1992)
Zhang, D.: Triangular norms on partially ordered sets. Fuzzy Sets and Systems 153(2), 195–209 (2005)
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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
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