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Eigen fuzzy number sets. (English) Zbl 0581.04007
Let X be a finite set and R:X\(\times X\to [0,1]\) be a fuzzy relation on X. A fuzzy set \(A: X\to [0,1]\) is an eigen fuzzy set associated with R if \(R\circ A=A\), where ”\(\circ ''\) stands for the max-min composition. If R is assigned and A is unknown, it suffices to apply the algorithms of E. Sanchez [Fuzzy Sets Syst. 1, 69-74 (1978; Zbl 0366.04001); J. Math. Anal. Appl. 81, 399-421 (1981; Zbl 0466.04003)] in order to find the greatest eigen fuzzy set associated with R. The authors solve a similar problem by replacing the entries in R and in A with fuzzy numbers. The definition of fuzzy number used by the authors [Fuzzy Sets Syst. 10, 87- 99 (1983; Zbl 0521.54001)] is a slight variant of that of D. Dubois and H. Prade [Int. J. Syst. Sci. 9, 613-626 (1978; Zbl 0383.94045)]. Interesting topological properties of particular fuzzy numbers, namely \(\theta\)-crisp, are also established.
Reviewer: S.Sessa

MSC:
03E20 Other classical set theory (including functions, relations, and set algebra)
03E72 Theory of fuzzy sets, etc.
54A40 Fuzzy topology
03B52 Fuzzy logic; logic of vagueness
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References:
[1] Dubois, D.; Prade, H., ()
[2] Dubois, D.; Prade, H., Operations on fuzzy numbers, Internat. J. systems sci., 9, 613-626, (1978) · Zbl 0383.94045
[3] Goetschel, R.; Voxman, W., Topological properties of fuzzy numbers, Fuzzy sets and systems, 10, 87-99, (1983) · Zbl 0521.54001
[4] Istratescu, V., ()
[5] Sanchez, E., Resolution of eigen fuzzy sets equations, Fuzzy sets and systems, 1, 69-75, (1978) · Zbl 0366.04001
[6] Sanchez, E., Eigen fuzzy sets and fuzzy relations, J. math. anal. appl., 81, 399-421, (1981) · Zbl 0466.04003
[7] Tashiro, T., Method of solution to inverse problem of fuzzy correspondence model, (), 70-79
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