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 |
References:
| [1] | Dubois, D.; Prade, H., (Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York) · Zbl 0444.94049 |
| [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., (Fixed Point Theory (1981), Reidel: Reidel Dordrecht) |
| [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, (Summary of Papers on General Fuzzy Problems (1980), Working Group Fuzzy Systems: Working Group Fuzzy Systems Tokyo), 70-79 |
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