zbMATH — the first resource for mathematics

Infinite fuzzy relation equations with continuous t-norms. (English) Zbl 1135.03346
Summary: This study develops a concept of infinite fuzzy relation equations with a continuous t-norm. It extends the work by Q. Xiong and X. Wang [“Some properties of sup-min fuzzy relational equations on infinite domains”, Fuzzy Sets Syst. 151, 393–402 (2005; Zbl 1062.03053)]. We describe attainable (respectively, unattainable) solutions, which are closely related to minimal solutions to the equations. It is shown that a solution set comprises both attainable and unattainable solutions. The study offers a characterization of these solutions. Under some assumptions, the solution set is presented and discussed. Two applications are also given.

03E72 Theory of fuzzy sets, etc.
Full Text: DOI
[1] Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Dordrecht · Zbl 0694.94025
[2] Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081
[3] Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Information sciences, 171, 145-172, (2005) · Zbl 1078.68815
[4] 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
[5] Mizumoto, M.; Zimmermann, H.J., Comparison of fuzzy reasoning methods, Fuzzy sets and systems, 8, 253-283, (1982) · Zbl 0501.03013
[6] Pedrycz, W., On generalized fuzzy relational equations and their applications, Journal of mathematical analysis and applications, 107, 520-536, (1985) · Zbl 0581.04003
[7] 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
[8] Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048
[9] Shieh, B.S., Solutions of fuzzy relation equations based on continuous t-norms, Information sciences, 177, 4208-4215, (2007) · Zbl 1122.03054
[10] 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
[11] 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
[12] Wang, X.P., Infinite fuzzy relational equations on a complete Brouwerian lattice, Fuzzy sets and systems, 138, 657-666, (2003) · Zbl 1075.03026
[13] Wang, X.P.; Qu, X.B., Continuous join-irreducible elements and their applications to describing the solution set of fuzzy relational equations, Acta Mathematica sinica, 49, 5, 1171-1180, (2006), (in Chinese) · Zbl 1120.03042
[14] Xiong, Q.Q.; Wang, X.P., Some properties of sup-MIN fuzzy relational equations on infinite domains, Fuzzy sets and systems, 151, 393-402, (2005) · Zbl 1062.03053
[15] Zadeh, L.A.; Desoer, C.A., Linear system theory, (1963), Mc. Graw-Hill New York
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.