Solutions of fuzzy relation equations based on continuous t-norms. (English) Zbl 1122.03054

Summary: This study is concerned with fuzzy relation equations with continuous t-norms in the form \(ATR = B\), where \(A\) and \(B\) are fuzzy subsets of \(X\) and \(Y\), respectively; \(R \subset X \times Y\) is a fuzzy relation, and \(T\) is a continuous t-norm. The problem is how to determine \(A\) from \(R\) and \(B\). First, an “if and only if” condition of being solvable is presented. Novel algorithms are then presented for determining minimal solutions when \(X\) and \(Y\) are finite. The proposed algorithms generate all minimal solutions for the equations, making them efficient solving procedures.


03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Baets, B.D., Analytical solution methods for fuzzy relational equations, (), 291-340 · Zbl 0970.03044
[2] Bandemer, H.; Gottwald, S., Fuzzy sets, fuzzy logic, fuzzy methods with applications, (1995), John Wiley & Sons Ltd. England · Zbl 0833.94028
[3] 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
[4] Chakraborty, M.K.; Das, M., Studies in fuzzy relations over fuzzy subsets, Fuzzy sets and systems, 9, 79-89, (1983) · Zbl 0519.04002
[5] Chen, L.; Wang, P.P., Fuzzy relation equation (II): the branch-point-solutions and the categorized minimal solutions, Soft computing, 11, 33-40, (2007) · Zbl 1108.03310
[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] Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy sets and systems, 7, 89-101, (1982) · Zbl 0483.04001
[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] Fernandez, M.J.; Gil, P., Some specific types of fuzzy relation equations, Information sciences, 164, 189-195, (2004) · Zbl 1058.03058
[10] Gavalec, M., Solvability and unique solvability of max – min fuzzy equations, Fuzzy sets and systems, 124, 385-393, (2001) · Zbl 0994.03047
[11] Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006
[12] Hirota, K.; Pedrycz, W., Solving fuzzy relational equations through logical filtering, Fuzzy sets and systems, 81, 355-363, (1996) · Zbl 0877.04005
[13] Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081
[14] Imai, H.; Miyakoshi, M.; Da-te, Tsutomu, Some properties of minimal solutions for a fuzzy relation equation, Fuzzy sets and systems, 90, 335-340, (1997) · Zbl 0919.04008
[15] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic theory and applications, (1995), Prentice Hall PTR, Pearson Education, Inc. · Zbl 0915.03001
[16] Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Information sciences, 171, 145-172, (2005) · Zbl 1078.68815
[17] Luoh, L.; Wang, W.J.; Liaw, Y.K., New algorithms for solving fuzzy relation equations, Mathematics and computers in simulation, 59, 329-333, (2002) · Zbl 0999.03513
[18] Markovskii, A.V., On the relation between equations with MAX-product composition and the covering problem, Fuzzy sets and systems, 153, 261-273, (2005) · Zbl 1073.03538
[19] Neundorf, D.; Bohm, R., Solvability criteria for systems of fuzzy relation equations, Fuzzy sets and systems, 80, 345-352, (1996)
[20] Pedrycz, W., Inverse problem in fuzzy relational equations, Fuzzy sets and systems, 36, 277-291, (1990) · Zbl 0708.04003
[21] Pedrycz, W., Processing in relational structures: fuzzy relational equations, Fuzzy sets and systems, 25, 77-106, (1991) · Zbl 0721.94030
[22] Pedrycz, W., S-t fuzzy relational equations, Fuzzy sets and systems, 59, 189-195, (1993)
[23] Perfilieva, I.; Novak, V., System of fuzzy relation equations as a continuous model of IF-THEN rules, Information sciences, 177, 3218-3227, (2007) · Zbl 1124.03029
[24] Perfilieva, I.; Tonis, A., Compatibility of systems of fuzzy relation equations, International journal of general systems, 29, 511-528, (2000) · Zbl 0955.03062
[25] Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048
[26] Stamou, G.B.; Tzafestas, S.G., Resolution of composite fuzzy relation equations based on Archimedean triangular norms, Fuzzy sets and systems, 120, 395-407, (2001) · Zbl 0979.03042
[27] Wang, X.P., Method of solution to fuzzy relation equations in a complete Brouwerian lattice, Fuzzy sets and systems, 120, 409-414, (2001) · Zbl 0981.03055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.