# zbMATH — the first resource for mathematics

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
##### Keywords:
eigen fuzzy set; fuzzy numbers
Full Text:
##### 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
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.