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Bivalent and other solutions of fuzzy relational equations via linguistic hedges. (English) Zbl 1258.03080
In the paper, it is shown that constrained solutions of ordinary fuzzy relational equations appear as solutions of a new type of fuzzy relational equations, i.e., equations which use modified compositions of fuzzy relations. The mentioned modification consists in using linguistic hedges in the description of the compositions. It is proved that the well-known results regarding solutions of fuzzy relational equations may be generalized to the more general case which involves intensifying or relaxing hedges. This idea emphasizes the role of linguistic hedges as constraints. Results are developed for complete residuated lattices as the structure of truth degrees. In the paper, both the intensifying (truth-stressing) hedges as “very” or “highly”, and the relaxing (truth-depressing) hedges such as “more or less” or “roughly” are applied. These hedges are understood as unary functions on the set of truth degrees. Moreover, an illustrative example is provided which demonstrates the results of the paper and the usefulness of constrained solutions of fuzzy relational equations.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 03B52 Fuzzy logic; logic of vagueness
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##### References:
 [1] Baaz, M., Infinite-valued Gödel logics with 0-1 projections and relativizations, (), 23-33 · Zbl 0862.03015 [2] Bandler, W.; Kohout, L.J., Semantics of implication operators and fuzzy relational products, Int. J. man-Mach. stud., 12, 89-116, (1980) · Zbl 0435.68042 [3] Bandler, W.; Kohout, L.J., Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems, (), 341-367 [4] Belohlavek, R.; Funioková, T.; Vychodil, V., Fuzzy closure operators with truth stressers, Logic J. IGPL, 13, 5, 503-513, (2005) · Zbl 1089.06001 [5] R. Belohlavek, V. Vychodil, Reducing the size of fuzzy concept lattices by hedges, in: Proceedings of FUZZ-IEEE, Reno, Nevada, 2005, pp. 663-668. [6] Belohlavek, R.; Vychodil, V., Attribute implications in a fuzzy setting, (), 45-60 · Zbl 1177.68203 [7] Belohlavek, R.; Vychodil, V., Fuzzy concept lattices constrained by hedges, J. adv. comput. intell. intell. inf., 11, 6, 536-545, (2007) [8] De Baets, B., Analytical solution methods for fuzzy relational equations, (), 291-340 · Zbl 0970.03044 [9] Di Nola, A.; Sanchez, E.; Pedrycz, W.; Sessa, S., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer · Zbl 0694.94025 [10] S. Gottwald, Fuzzy Sets and Fuzzy Logic. Foundations of Applications—From a Mathematical Point of View, Vieweg, Wiesbaden, 1993. · Zbl 0782.94025 [11] Gottwald, S., A treatise on many-valued logics, (2001), Research Studies Press Baldock, Hertfordshire, England · Zbl 1048.03002 [12] Gottwald, S., Calculi of information granules: fuzzy relational equations, (), doi: 10.1002/9780470724163.ch11 [13] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030 [14] Hájek, P., On very true, Fuzzy sets syst., 124, 329-333, (2001) · Zbl 0997.03028 [15] Hirota, K.; Pedrycz, W., Solving fuzzy relational equations through logical filtering, Fuzzy sets syst., 81, 355-363, (1996) · Zbl 0877.04005 [16] Hirota, K.; Pedrycz, W., Specificity shift in solving fuzzy relational equations, Fuzzy sets syst., 106, 211-220, (1999) [17] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic. theory and applications, (1995), Prentice-Hall · Zbl 0915.03001 [18] Sanchez, E., Resolution of composite fuzzy relation equations, Inf. control, 30, 1, 38-48, (1976) · Zbl 0326.02048 [19] Takeuti, G.; Titani, S., Globalization of intuitionistic set theory, Ann. pure appl. logic, 33, 195-211, (1987) · Zbl 0633.03050 [20] Vychodil, V., Truth-depressing hedges and BL-logic, Fuzzy sets syst., 157, 2074-2090, (2006) · Zbl 1114.03023 [21] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning—I, Inf. sci., 8, 3, 199-249, (1975) · Zbl 0397.68071
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