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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.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
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##### References:
  Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Dordrecht · Zbl 0694.94025  Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081  Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Information sciences, 171, 145-172, (2005) · Zbl 1078.68815  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  Mizumoto, M.; Zimmermann, H.J., Comparison of fuzzy reasoning methods, Fuzzy sets and systems, 8, 253-283, (1982) · Zbl 0501.03013  Pedrycz, W., On generalized fuzzy relational equations and their applications, Journal of mathematical analysis and applications, 107, 520-536, (1985) · Zbl 0581.04003  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  Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048  Shieh, B.S., Solutions of fuzzy relation equations based on continuous t-norms, Information sciences, 177, 4208-4215, (2007) · Zbl 1122.03054  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  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  Wang, X.P., Infinite fuzzy relational equations on a complete Brouwerian lattice, Fuzzy sets and systems, 138, 657-666, (2003) · Zbl 1075.03026  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  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  Zadeh, L.A.; Desoer, C.A., Linear system theory, (1963), Mc. Graw-Hill New York
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