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New trends on the permutability equation. (English) Zbl 1305.39017

Summary: We study several functional equations, in two variables, that are closely related to the functional equation of permutability. We analyze the hierarchy among these equations, paying a particular attention to their use in fuzzy set theory.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B05 General theory of functional equations and inequalities
03E72 Theory of fuzzy sets, etc.
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[1] Aczél J.: Lectures on Functional Equations and their Applications. Academic Press, New York (1966) · Zbl 0139.09301
[2] Aczél J.: A Short Course on Functional Equations Based upon Recent Applications to the Social and Behavioral Sciences. D. Reidel, Dordrecht (1987) · Zbl 0607.39002
[3] Aczél J.: The associativity equation re-revisited. AIP Conf. Proc. 707, 195-203 (2004)
[4] Aczél J., Belousov V.D., Hosszú M.: Generalized associativity and bisymmetry on quasigroups. Acta Mathematica Academiae Scientiarum Hungaricae 11, 127-136 (1960) · Zbl 0090.24301
[5] Aczél J., Dhombres J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989) · Zbl 0685.39006
[6] Alsina C., Frank M.J., Schweizer B.: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Co. Pte. Ltd., Singapore (2006) · Zbl 1100.39023
[7] Baczyński M., Jayaram B.: Fuzzy Implications. Studies in Fuzzines and Soft Computing, vol. 231. Springer, Berlin (2008) · Zbl 1147.03012
[8] Beliakov G., Bustince H., Fernández J.: The median and its extensions. Fuzzy Sets Syst. 175(1), 36-47 (2011) · Zbl 1217.62012
[9] Beliakov, G., Calvo, T.: On migrative means and copulas. In: Proceedings of the Fifth International Summer School on Aggregation Operators. Palma de Mallorca, UK, pp. 107-110 (2009)
[10] Bustince H., Burillo P., Soria F.: Automorphisms, negations and implication operators. Fuzzy Sets Syst. 134, 209-219 (2003) · Zbl 1010.03017
[11] Bustince H., De Baets B., Fernández J., Mesiar R., Montero J.: A generalization of the migrativity property of aggregation functions. Inf. Sci. 191, 76-85 (2012) · Zbl 1248.62003
[12] Bustince H., Fernández J., Mesiar R., Montero J., Orduna R.: Overlap functions. Nonlinear Anal. 72, 1488-1499 (2010) · Zbl 1182.26076
[13] Bustince H., Fernández J., Sanz J., Baczyński M., Mesiar R.: Construction of strong equality index from implication operators. Fuzzy Sets Syst. 211, 15-33 (2013) · Zbl 1267.03038
[14] Bustince H., Montero J., Mesiar R.: Migrativity of aggregation functions. Fuzzy Sets Syst. 160, 766-777 (2009) · Zbl 1186.68459
[15] Campión M.J., Candeal J.C., Catalán R.G., De Miguel J.R., Induráin E., Molina J.A.: Aggregation of preferences in crisp and fuzzy settings: functional equations leading to possibility results. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 19, 89-114 (2011) · Zbl 1214.91030
[16] Candeal J.C., De Miguel J.R., Induráin E., Olóriz E.: Associativity equation revisited. Publ. Math. Debr. 51, 133-144 (1997) · Zbl 0889.39019
[17] Candeal J.C., Induráin E.: Bivariate functional equations around associativity. Aequationes Math. 84, 137-155 (2012) · Zbl 1271.39015
[18] Castillo-Ron E., Ruiz-Cobo R.: Functional Equations in Science and Engineering. Marcel Dekker, New York (1992) · Zbl 0801.39004
[19] Craigen R., Páles Z.: The associativity equation revisited. Aequationes Math. 37, 306-312 (1989) · Zbl 0676.39005
[20] Durante F., Sarkoci P.: A note on the convex combinations of triangular norms. Fuzzy Sets Syst. 159, 77-80 (2008) · Zbl 1173.03042
[21] Falmagne J.-C., Doignon J.-P.: Axiomatic derivation of the Doppler factor and related relativistic laws. Aequationes Math. 80, 85-99 (2010) · Zbl 1247.39017
[22] Falmagne, J.-C.: On a Class of Meaningful Permutable Laws. University of California, Irvine (2012, preprint) · Zbl 0889.39019
[23] Fodor J., Rudas I.J.: On continuous triangular norms that are migrative. Fuzzy Sets Syst. 158, 1692-1697 (2007) · Zbl 1120.03035
[24] Gorenstein D.: The Enormous Theorem. Sci. Am. 253, 104-115 (1985)
[25] Grabisch M., Marichal J.-L., Mesiar R., Pap E. (2009) Aggregation functions. Encyclopedia of Mathematics and its Applications, vol. 127, Cambridge University Press, UK · Zbl 1182.26076
[26] Hosszú M.: Note on commutable mappings. Publ. Math. Debr. 9, 105-106 (1962) · Zbl 0186.47703
[27] Klement E.P., Mesiar R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0972.03002
[28] Kuczma M.: Functional Equations in a Single Variable. Polska Akademia Nauk, Warsaw (1968) · Zbl 0196.16403
[29] Ling C.H.: Representation of associative functions. Publ. Math. Debr. 12, 189-212 (1965) · Zbl 0137.26401
[30] López-Molina, C., De Baets, B., Bustince, H., Induráin, E., Stupñanová, A., Mesiar R.: Bimigrativity of binary aggregation functions (2013, in review) · Zbl 1341.68262
[31] Marichal J.-L.: On the associativity functional equation. Fuzzy Sets Syst. 114, 381-389 (2000) · Zbl 0962.39012
[32] Marichal J.-L.: Solving Chisini’s functional equation. Aequationes Math. 79(3), 237-260 (2010) · Zbl 1214.26017
[33] Marichal J.-L., Mathonet P.: A description of n-ary semigroups polynomial-derived from integral domains. Semigroup Forum 83(2), 241-249 (2011) · Zbl 1245.20072
[34] Mesiar R., Bustince H., Fernández J.: On the α-migrativity of semicopulas, quasi-copulas, and copulas. Inf. Sci. 180, 1967-1976 (2010) · Zbl 1191.62096
[35] Montero, J., Gómez, D., Muñoz, S.: Fuzzy information representation for decision aiding. In: Proceedings of the IPMU Conference, Málaga, Spain, pp. 1425-1430 (2008)
[36] Montero, J., López, V., Gómez, D. et al.: The role of fuzziness in decision making. In: Ruan, D. Fuzzy Logic: A Spectrum of Applied and Theoretical Issues, pp. 337-349. Springer, Berlin (2007) · Zbl 0090.24301
[37] Roy B.: Decision sciences or decision aid sciences. Eur. J. Oper. Res. 66, 184-203 (1993)
[38] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York (1983) · Zbl 0546.60010
[39] Small C.G.: Functional Equations and How to Solve Them. Springer, New York (2007) · Zbl 1152.39300
[40] Zadeh L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man Cybern. 9, 28-44 (1973) · Zbl 0273.93002
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