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Semiautoduality in a restricted family of aggregation operators. (English) Zbl 1123.68125

Summary: We consider aggregation operators satisfying non-decreasingness and some specific boundary conditions. We then analyze some properties of such a family of aggregation operators, introducing the semiautoduality condition, which is weaker than the standard autoduality condition (i.e., the standard self De Morgan identity). Particular families of aggregation operators will appear depending on the context.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
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[1] Aczél, J., On mean values, Bull. Amer. Math. Soc., 54, 392-400 (1948) · Zbl 0030.02702
[2] Alsina, C.; Mayor, G.; Tomas, M. S.; Torrens, J., A characterization of a class of aggregation functions, Fuzzy Sets and Systems, 53, 33-88 (1993) · Zbl 0788.04004
[3] Amo, A.; Montero, J.; Molina, E., Representation of consistent recursive rules, European J. Oper. Res., 130, 29-53 (2001) · Zbl 1137.03322
[4] Bustince, H.; Mohedano, V.; Barrenechea, E.; Pagola, M., Definition and construction of fuzzy DI-subsethood measures, Inform. Sci., 176, 3190-3231 (2006) · Zbl 1104.03052
[5] Calvo, T.; Kolesarova, A.; Komornikova, M.; Mesiar, R., Aggregation operators: properties, classes and construction methods, (Calvo, T.; Mayor, G.; Mesiar, R., Aggregation Operators (2002), Springer: Springer Berlin), 3-104 · Zbl 1039.03015
[6] Calvo, T.; Mesiar, R., Weighted triangular norms-based aggregation operators, Fuzzy Sets and Systems, 137, 3-10 (2003) · Zbl 1022.03035
[7] Calvo, T.; Mesiar, R.; Yager, R. R., Quantitative weights and aggregation, IEEE Trans. Fuzzy Systems, 12, 62-69 (2004)
[8] Chiclana, F.; Herrera, F.; Herrera-Viedma, E.; Martínez, L., A note on the reciprocity in the aggregation of fuzzy preference relations using OWA operators, Fuzzy Sets and Systems, 137, 71-83 (2003) · Zbl 1056.91016
[9] Cholewa, W., Aggregation for fuzzy opinions—an axiomatic approach, Fuzzy Sets and Systems, 17, 249-258 (1985) · Zbl 0597.90005
[10] Cutello, V.; Montero, J., Recursive connective rules, Internat. J. Intelligent Systems, 14, 3-20 (1999) · Zbl 0955.68103
[11] Dombi, J., A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets and Systems, 8, 149-163 (1982) · Zbl 0494.04005
[12] Dombi, J., Basic concept for a theory of evaluation: the aggregative operator, European J. Oper. Res., 10, 282-293 (1982) · Zbl 0488.90003
[13] Dubois, D.; Koning, J. L., Social choice axioms for fuzzy set aggregation, Fuzzy Sets and Systems, 58, 339-342 (1991)
[14] Dubois, D.; Prade, H., A class of fuzzy measures based on triangular norms, Internat. J. Gen. Systems, 8, 43-61 (1982) · Zbl 0473.94023
[15] Dubois, D.; Prade, H., Criteria aggregation and ranking of alternatives in the framework of fuzzy theory, (Zimmermann, H. J.; Zadeh, L. A.; Gaines, B., Fuzzy Sets and Decision Analysis, TIMS Studies in Management Science, Vol. 20 (1984)), 209-240
[16] Dubois, D.; Prade, H., A review of fuzzy ser aggregation connectives, Inform. Sci., 36, 85-121 (1985) · Zbl 0582.03040
[17] Dujmovic, J. J., Weighted conjunctive and disjunctive means and their application in system evaluation, Univ. Beograd. Publ. Elektrotechn. Fak., 147-158 (1974) · Zbl 0297.65012
[18] Esteva, F.; Trillas, E.; Domingo, X., Weak and strong negation function for fuzzy set theory, (Proc. 11th IEEE Internat. Symp. on Multivalued Logic (1981), Norman: Norman Oklahoma), 23-27 · Zbl 0548.03036
[19] Fodor, J.; Roubens, M., Fuzzy Preference Modelling and Multicriteria Decision Support (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0827.90002
[20] Frank, M. J., On the simultaneous associativity of \(F(x, y)\) and \(x + y - F(x, y)\), Aequationes Math., 19, 194-226 (1979) · Zbl 0444.39003
[21] Fung, L. W.; Fu, K. S., An axiomatic approach to rational decision making in a fuzzy environment, (Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Simura, M., Fuzzy Sets and their Applications to Cognitive and Decision Processes (1975), Academic Press: Academic Press New York), 227-256 · Zbl 0366.90003
[22] Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174 (1967) · Zbl 0145.24404
[23] Goguen, J. A., The logic of inexact concepts, Synthese, 19, 325-373 (1969) · Zbl 0184.00903
[24] Gómez, D.; Montero, J., A discussion on aggregation operators, Kybernetika, 40, 107-120 (2004) · Zbl 1249.68229
[25] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1988), Cambridge Mathematical Library · Zbl 0634.26008
[26] Klement, E. P.; Mesiar, R.; Pap, E., On the relationship of associative compensatory operators to triangular norms and conorms, Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems, 4, 129-144 (1996) · Zbl 1232.03041
[27] Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0972.03002
[28] Klir, G.; Yuan, B., Fuzzy Sets and Fuzzy Logic: Theory and Applications (1995), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001
[29] Klir, G. J.; Folger, T. A., Fuzzy Sets Uncertainty and Information (1988), Prentice-Hall International · Zbl 0675.94025
[30] Kolesárová, A., Limit properties of quasi-arithmetic means, Fuzzy Sets and Systems, 124, 65-71 (2001) · Zbl 0989.03060
[31] Kolesárová, A.; Komorníková, M., Triangular norm-based iterative compensatory operators, Fuzzy Sets and Systems, 104, 109-120 (1999) · Zbl 0931.68123
[32] Kolmogoroff, A. N., Sur la notion de la moyenne, Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez., 12, 388-391 (1930) · JFM 56.0198.02
[33] Lowen, R., On fuzzy complements, Inform. Sci., 14, 107-113 (1978) · Zbl 0416.03047
[34] J.L. Marichal, Aggregation operators for multicriteria decision aid, Ph.D. Thesis, University of Liege, 1998.; J.L. Marichal, Aggregation operators for multicriteria decision aid, Ph.D. Thesis, University of Liege, 1998.
[35] Mayor, G.; Torrens, J., On a class of binary operations: non-strict Archimedian aggregation functions, (Proc. 18th ISMVL, Palma de Mallorca (1988)), 54-59
[36] Mayor, G.; Trillas, E., On the representation of some aggregation functions, (Proc. 16th ISMVL, Blacksburg (1986)), 110-114
[37] Mesiar, R.; Komornikova, M., Aggregation operators, (Herceg, D.; Surla, K., Proc. PRIM Conf. on Applied Mathematics. Proc. PRIM Conf. on Applied Mathematics, Budva (1996)), 193-211 · Zbl 0960.03045
[38] Montero, J., A note of Fung-Fu’s theorem, Fuzzy Sets and Systems, 13, 259-269 (1985) · Zbl 0606.90008
[39] Montero, J., Rational aggregation rules, Fuzzy Sets and Systems, 62, 267-276 (1994) · Zbl 0828.90005
[40] Nagumo, M., Uber eine Klasse der Mittelwerte, Japanese J. Math., 6, 71-79 (1930) · JFM 56.0198.03
[41] Ovchinnikov, S. V., Structure of fuzzy binary relations, Fuzzy Sets and Systems, 6, 2, 169-195 (1981) · Zbl 0464.04004
[42] Ovchinnikov, S. V., General negations in fuzzy set theory, J. Math. Anal. Appl., 92, 234-239 (1983) · Zbl 0518.04003
[43] Ovchinnikov, S. V.; Roubens, M., On strict preference relations, Fuzzy Sets and Systems, 43, 319-326 (1991) · Zbl 0747.90006
[44] Ramakrishnan, R.; Rao, C. J.M., The fuzzy weighted additive rule, Fuzzy Sets and Systems, 46, 177-187 (1992) · Zbl 0768.90043
[45] Silvert, W., Symmetric summation: a class of operations on fuzzy subsets, IEEE Trans. Systems Man Cybernet., 9, 659-667 (1979) · Zbl 0424.04003
[46] Sinha, D.; Dougherty, E. R., Fuzzification of set inclusion: theory and applications, Fuzzy Sets and Systems, 55, 15-42 (1993) · Zbl 0788.04007
[47] Torra, V., On some aggregation operators for numerical information, (Torra, V., Information Fusion in Data Mining (2003), Springer: Springer Berlin) · Zbl 1095.68696
[48] E. Trillas, Sobre funciones de negación en la teoría de conjuntos difusos, Stochastica III-1 (1979) 47-59 (in Spanish) (Reprinted (English version), in: S. Barro, A. Sobrino, A. Bugarin (Eds.), Advances of Fuzzy Logic, Universidad de Santiago de Compostela, 1998, pp. 31-43).; E. Trillas, Sobre funciones de negación en la teoría de conjuntos difusos, Stochastica III-1 (1979) 47-59 (in Spanish) (Reprinted (English version), in: S. Barro, A. Sobrino, A. Bugarin (Eds.), Advances of Fuzzy Logic, Universidad de Santiago de Compostela, 1998, pp. 31-43).
[49] E. Trillas, C. Alsina, J.M. Terricabras, Introducción a la lógica borrosa. Ariel Matemática (1995).; E. Trillas, C. Alsina, J.M. Terricabras, Introducción a la lógica borrosa. Ariel Matemática (1995).
[50] Yager, R. R., On the measure of fuzziness and negation. Part I : membership in the unit interval, Internat. J. Gen. Systems, 5, 221-229 (1979) · Zbl 0429.04007
[51] Yager, R. R., On the measure of fuzziness and negation. Part II: lattices, Inform. Control, 44, 236-260 (1979) · Zbl 0429.04008
[52] Yager, R. R., Families of OWA operators, Fuzzy Sets and Systems, 59, 125-148 (1993) · Zbl 0790.94004
[53] Yager, R. R., Induced aggregation operators, Fuzzy Sets and Systems, 137, 59-69 (2003) · Zbl 1056.68146
[54] Young, V. R., Fuzzy subsethood, Fuzzy Sets and Systems, 77, 371-384 (1996) · Zbl 0872.94062
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