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Resolution of composite fuzzy relation equations based on Archimedean triangular norms. (English) Zbl 0979.03042
The paper investigates the existence of a solution for composite fuzzy relation equations in any t-norm $$t$$ and proposes a method for solving sup-$$t$$-norm fuzzy relation equations with an Archimedean t-norm $$t$$. The numerical methods are simpler and faster than those developed for all continuous t-norms. This improvement is essential as the authors show that the only continuous non-Archimedean t-norm is the “minimum”.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
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##### References:
 [1] Akiyama, Y.; Abe, T.; Mitsunaga, T.; Koga, H., A conceptual study MAX-$$\^{}\{*\}$$ composition on the correspondence of base spaces and its applications in determining fuzzy relations, Jpn. J. fuzzy theory systems, 3, 2, 113-132, (1991) · Zbl 0800.04011 [2] Blanco, A.; Delgado, M.; Requena, I., Improved fuzzy neural networks for solving relational equations, Fuzzy sets and systems, 72, 311-322, (1995) [3] Bourke, M.; Fisher, D.G., Calculation and application of a minimum fuzzy relational matrix, Fuzzy sets and systems, 74, 225-236, (1995) · Zbl 0937.93026 [4] Bourke, M.; Fisher, D.G., Solution algorithms for fuzzy relational equations with MAX-product composition, Fuzzy sets and systems, 94, 61-69, (1998) · Zbl 0923.04003 [5] Cheng, L.; Peng, B., The fuzzy relation equation with union or intersection preserving operator, Fuzzy sets and systems, 25, 191-204, (1988) · Zbl 0651.04005 [6] Chung, F.; Lee, T., A new look at solving a system of fuzzy relational equations, Fuzzy sets and systems, 88, 343-353, (1997) · Zbl 0914.04002 [7] De Oliveira, J.-V., Neuron inspired learning rules for fuzzy relational structures, Fuzzy sets and systems, 57, 41-53, (1993) [8] Di Nola, A.; Pedrycz, W.; Sessa, S.; Sanchez, E., Fuzzy relation equations and their application to knowledge engineering, (1989), Kluwer Academic Press Dordrecht [9] Di Nola, A.; Pedrycz, W.; Sessa, S.; Sanchez, E., Fuzzy relation equations theory as a basis of fuzzy modellingan overview, Fuzzy sets and systems, 40, 415-429, (1991) · Zbl 0727.04005 [10] Gottwald, S., Approximately solving fuzzy relation equationssome mathematical results and some heuristic proposals, Fuzzy sets and systems, 66, 175-193, (1994) · Zbl 0842.04010 [11] Gupta, M.M.; Qi, J., Theory of T-norms and fuzzy inference, Fuzzy sets and systems, 40, 431-450, (1991) · Zbl 0726.03017 [12] Gupta, M.M.; Qi, J., Design of fuzzy logic controllers based on generalized T-operators, Fuzzy sets and systems, 40, 473-489, (1991) · Zbl 0732.93050 [13] Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006 [14] Hirota, K.; Pedrycz, W., Solving fuzzy relational equations through logical filtering, Fuzzy sets and systems, 81, 355-363, (1996) · Zbl 0877.04005 [15] Hong, D.H.; Hwang, S.Y., On the compositional rule of inference under triangular norms, Fuzzy sets and systems, 66, 25-38, (1994) · Zbl 1018.03511 [16] Ikoma, N.; Pedrycz, W.; Hirota, K., Estimation of fuzzy relational matrix by using probabilistic descent method, Fuzzy sets and systems, 57, 335-349, (1993) [17] Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081 [18] Jenei, S., On Archimedean triangular norms, Fuzzy sets and systems, 99, 179-186, (1998) · Zbl 0938.03083 [19] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logictheory and applications, (1995), Prentice-Hall, PTR USA [20] Li, J.X., On an algorithm for solving fuzzy linear systems, Fuzzy sets and systems, 61, 369-371, (1994) · Zbl 0826.04004 [21] Pappis, C.P.; Adamopoulos, G.I., A computer algorithm for the solution of the inverse problem of fuzzy systems, Fuzzy sets and systems, 39, 279-290, (1991) · Zbl 0727.93029 [22] Pappis, C.P.; Sugeno, M., Fuzzy relation equations and the inverse problem, Fuzzy sets and systems, 15, 79-90, (1985) · Zbl 0561.04003 [23] Pedrycz, W., Fuzzy relational equations with generalized connectives and their applications, Fuzzy sets and systems, 10, 185-201, (1983) · Zbl 0525.04004 [24] Pedrycz, W., Approximate solutions of fuzzy relational equations, Fuzzy sets and systems, 28, 183-202, (1988) · Zbl 0669.04002 [25] Pedrycz, W., Direct and inverse problem in comparison of fuzzy data, Fuzzy sets and systems, 34, 223-235, (1990) [26] Pedrycz, W., Inverse problem in fuzzy relational equations, Fuzzy sets and systems, 36, 277-291, (1990) · Zbl 0708.04003 [27] Pedrycz, W., Neurocomputations in relational systems, IEEE trans. pattern anal. Mach. intell., 13, 3, 289-297, (1991) [28] Pedrycz, W., Processing in relational structuresfuzzy relational equations, Fuzzy sets and systems, 25, 77-106, (1991) · Zbl 0721.94030 [29] Pedrycz, W., S-t fuzzy relational equations, Fuzzy sets and systems, 59, 189-195, (1993) [30] Pedrycz, W., Genetic algorithms for learning in fuzzy relational structures, Fuzzy sets and systems, 69, 37-52, (1995) [31] Peeva, K., Fuzzy linear systems, Fuzzy sets and systems, 49, 339-355, (1992) · Zbl 0805.04005 [32] Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048 [33] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. math. deprecen, 10, 69-81, (1963) · Zbl 0119.14001 [34] Wagenknecht, M.; Hartmann, K., On the existence of minimal solutions for fuzzy equations with tolerances, Fuzzy sets and systems, 34, 237-244, (1990) · Zbl 0687.90094 [35] Wu, W., Fuzzy reasoning and fuzzy relational equations, Fuzzy sets and systems, 20, 67-78, (1986) · Zbl 0629.94031 [36] Zeng, X.J.; Singh, M.G., Approximation theory of fuzzy systems - SISO case, IEEE trans. fuzzy systems, 2, 2, 162-176, (1994) [37] Zeng, X.J.; Singh, M.G., Approximation theory of fuzzy systems - MIMO case, IEEE trans. fuzzy systems, 3, 2, 219-235, (1995)
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