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Problems on triangular norms and related operators. (English) Zbl 1050.03019

Summary: A number of open problems on triangular norms and related operators was posed during the 24th Linz seminar on fuzzy set theory, “Triangular norms and related operators in many-valued logics”, held in February 2003. They are collected here, together with some other open problems in this context and with some problems which were posed earlier and have been solved in the meantime.

MSC:

03B52 Fuzzy logic; logic of vagueness
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[1] Aczél, J., Lectures on Functional Equations and their Applications (1966), Academic Press: Academic Press New York · Zbl 0139.09301
[2] C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math., in press.; C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math., in press. · Zbl 1077.39021
[3] Alsina, C.; Trillas, E., On the functional equation \(S_1(x,y)=S_2(x,T(N(x),y))\), (Daróczy, Z.; Páles, Z., Functional Equations, Results and Advances (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 323-334 · Zbl 0996.39021
[4] Benferhat, B.; Duboils, D.; Prade, H., From semantic to syntactic approaches to information combination in possibilistic logic, (Bouchon-Meunier, B., Aggregation and Fusion of Imperfect Information, Physica (1997), Verlag: Verlag Heidelberg), 141-161 · Zbl 0898.68079
[5] Bezdek, J. C.; Harris, J. D., Fuzzy partitions and relationsan axiomatic basis for clustering, Fuzzy Sets and Systems, 1, 111-127 (1978) · Zbl 0442.68093
[6] Bezdek, J. C.; Harris, J. D., Convex decompositions of fuzzy partitions, J. Math. Anal. Appl., 67, 490-512 (1979) · Zbl 0411.68056
[7] Budinčević, M.; Kurilić, M. S., A family of strict and discontinuous triangular norms, Fuzzy Sets and Systems, 95, 381-384 (1998) · Zbl 0922.04006
[8] De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy Sets and Systems, 104, 61-75 (1999) · Zbl 0935.03060
[9] Dubois, D.; Prade, H., Representation and combination of uncertainty with belief functions and possibility measures, Comput. Intell., 4, 244-264 (1988)
[10] Foder, J. C.; Yager, R. R.; Rybalov, A., Structure of uninorms, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 5, 411-427 (1997) · Zbl 1232.03015
[11] 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
[12] Fuchs, L., Partially Ordered Algebraic Systems (1963), Pergamon Press: Pergamon Press Oxford · Zbl 0137.02001
[13] Hadžić, O.; Pap, E., Fixed Point Theory in Probabilistic Metric Spaces (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[14] Hájek, P., Metamathematics of Fuzzy Logic (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[15] P. Hájek, 10,000 open problems in mathematical fuzzy logic, Personal communication, 2002 (available via e-mail: hajek@cs.cas.cz).; P. Hájek, 10,000 open problems in mathematical fuzzy logic, Personal communication, 2002 (available via e-mail: hajek@cs.cas.cz).
[16] Hájek, P., Observations on the monoidal t-norm logic, Fuzzy Sets and Systems, 132, 107-112 (2002) · Zbl 1012.03035
[17] Hájek, P.; Havaánek, T.; Jiroušek, R., Uncertain Information Processing in Expert Systems (1992), CRC Press: CRC Press Boca Raton
[18] S. Janssens, B. De Baets, H. De Meyer, Bell-type inequalities for triangular norms, 2003, submitted for publication.; S. Janssens, B. De Baets, H. De Meyer, Bell-type inequalities for triangular norms, 2003, submitted for publication. · Zbl 1249.54015
[19] S. Jenei, B. De Baets, On the direct decomposability of t-norms over direct product lattices, Fuzzy Sets and Systems, 2003, in press.; S. Jenei, B. De Baets, On the direct decomposability of t-norms over direct product lattices, Fuzzy Sets and Systems, 2003, in press. · Zbl 1032.03022
[20] Kasumov, N., Metric properties of fuzzy partitions, Fuzzy Sets and Systems, 81, 365-378 (1996) · Zbl 0885.62071
[21] Kimberling, C., On a class of associative functions, Publ. Math. Debrecen, 20, 21-39 (1973) · Zbl 0276.26011
[22] Klement, E. P., Construction of fuzzy \(σ\)-algebras using triangular norms, J. Math. Anal. Appl., 85, 543-565 (1982) · Zbl 0491.28003
[23] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic. Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.; E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic. Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dordrecht, 2000. · Zbl 0972.03002
[24] E.P. Klement, R. Mesiar, E. Pap, Archimedean components of trianglular norms, J. Austral. Math. Soc., in press, FLLL-TR-0206.; E.P. Klement, R. Mesiar, E. Pap, Archimedean components of trianglular norms, J. Austral. Math. Soc., in press, FLLL-TR-0206. · Zbl 1087.20041
[25] Kolesárová, A., A note on Archimedean triangular norms, BUSEFAL, 80, 57-60 (1999)
[26] G.M. Krause, Personal communication, 1999.; G.M. Krause, Personal communication, 1999.
[27] Ling, C. M., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401
[28] Mayor, G.; Torrens, J., On a class of operators for expert systems, Internat. J. Intell. Systems, 8, 771-778 (1993) · Zbl 0785.68087
[29] Mesiar, R.; Novák, V., Open problems from the 2nd Internt. Conf. on Fuzzy Sets Theory and its applications, Fuzzy Sets and Systems, 81, 185-190 (1996) · Zbl 0877.04003
[30] Mesiarová, A., Wild t-norms, J. Electrical Eng., 51, 12/s, 36-40 (2000) · Zbl 0971.26015
[31] Mesiarová, A., Continuous diagonals of triangular norms, J. Electrical Eng., 52, 10/s, 7-11 (2001) · Zbl 1044.03536
[32] Moynihan, R., On the class of \(T_T\) semigroups of probability distribution functions, Aequationes Math., 12, 249-261 (1975) · Zbl 0309.60013
[33] Saminger, S.; Mesiar, R.; Bodenhofer, U., Domination of aggregation operators and preservation of transitivity, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 10/s, 11-35 (2002) · Zbl 1053.03514
[34] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. Math., 10, 313-334 (1960) · Zbl 0091.29801
[35] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland New York · Zbl 0546.60010
[36] Šerstnev, A. N., Random normed spacesproblems of completeness, Kazan. Gos. Univ. Učen. Zap., 122, 3-20 (1962) · Zbl 0178.52404
[37] Smutná, D., On a peculiar t-norm, BUSEFAL, 75, 60-67 (1998)
[38] Smutná, D., Non-continuous t-norms with continuous diagonal, J. Electrical Eng., 51, 12/s, 51-53 (2000) · Zbl 0972.03026
[39] Vicenı́k, P., A note on generators of t-norms, BUSEFAL, 75, 33-38 (1998)
[40] Zagordny, D., The cancellation law for inf-convolution of convex functions, Studia Math., 110, 3, 271-282 (1994) · Zbl 0811.49012
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