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Solvability and unique solvability of max-min fuzzy equations. (English) Zbl 0994.03047
Summary: The problem of solvability and the problem of unique solvability of a fuzzy relation equation in an arbitrary max-min algebra are considered and corresponding necessary and sufficient conditions are presented. The results allow to solve both problems by an \(O(mnp)\) algorithm, where \(m\), \(n\), \(p\) are the dimensions of the corresponding relations in the equation. The existence of the greatest and of the least solution is also considered.

03E72 Theory of fuzzy sets, etc.
15B33 Matrices over special rings (quaternions, finite fields, etc.)
08A72 Fuzzy algebraic structures
15A06 Linear equations (linear algebraic aspects)
Full Text: DOI
[1] Butkovič, P., Strong regularity of matrices—a survey of results, Discrete appl. math., 48, 45-68, (1994) · Zbl 0804.06017
[2] Butkovič, P.; Cechlárová, K.; Szabó, P., Strong linear independence in bottleneck algebra, Linear algebra appl., 94, 133-155, (1987) · Zbl 0629.90093
[3] Cechlárová, K., Strong regularity of matrices in a discrete bottleneck algebra, Linear algebra appl., 128, 35-50, (1990) · Zbl 0704.15003
[4] Cechlárová, K., Unique solvability of max – min fuzzy equations and strong regularity of matrices over fuzzy algebra, Fuzzy sets and systems, 75, 165-177, (1995) · Zbl 0852.15011
[5] Cechlárová, K.; Kolesár, K., Strong regularity of matrices in a discrete bounded bottleneck algebra, Linear algebra appl., 256, 141-152, (1997) · Zbl 0877.15018
[6] Cechlárová, K.; Plávka, J., Linear independence in bottleneck algebras, Fuzzy sets and systems, 77, 337-348, (1996) · Zbl 0877.15017
[7] Cuninghame-Green, R.A., Minimax algebra, lecture notes in economics and mathematical systems, vol. 166, (1979), Springer Berlin
[8] Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Dordrecht · Zbl 0694.94025
[9] M. Gavalec, J. Plávka, Strong regularity of matrices in general max – min algebra, Linear Algebra Appl., submitted. · Zbl 1030.15016
[10] Gottwald, S., Fuzzy sets and fuzzy logic, (1993), Vieweg Verlag Braunschweig - Wiesbaden - Toulouse
[11] Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081
[12] Imai, H.; Miyakoshi, M.; Da-te, Ts., Some properties of minimal solutions for a fuzzy relation equation, Fuzzy sets and systems, 90, 335-340, (1997) · Zbl 0919.04008
[13] Pedrycz, W., Identification in fuzzy systems, IEEE trans. systems man cybernet., 14, 361-366, (1984) · Zbl 0551.93015
[14] Pedrycz, W., Fuzzy control and fuzzy systems, (1989), Taunton New York · Zbl 0800.68750
[15] I. Perfilieva, A. Tonis, Consistency of models for approximate reasoning based on fuzzy logic, Proc. 4th Europ. Congress IT & Soft Computing, Aachen, 1996, pp. 651-655.
[16] Perfilieva, I.; Tonis, A., Compatibility of systems of fuzzy relation equations, Internal J. general systems, 29, 511-528, (2000) · Zbl 0955.03062
[17] Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048
[18] K. Zimmermann, Extremal Algebra, Ekon. Ústav ČSAV, Praha, 1976 (in Czech).
[19] Zimmermann, V., Linear and combinatorial optimization in ordered algebraic structures, annals of discrete mathematics, vol. 10, (1981), North-Holland Amsterdam
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