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Shape preserving additions of fuzzy intervals. (English) Zbl 0921.04002
The paper deals with addition of fuzzy numbers based on triangular norms [cf. D. Dubois and H. Prade, IEEE Trans. Autom. Control 26, 926-936 (1981); R. Fullér and T. Keresztfalvi, Fuzzy Sets Syst. 51, 155-159 (1992); A. Marková, Fuzzy Sets Syst. 85, 379-384 (1997; Zbl 0904.04007)]. It brings a negative answer to a general question about invariant shapes under such addition and presents some special cases of shapes and triangular norms with addition preserving such shapes. E.g., addition with the boundary triangular norms (minimum product and drastic product) preserves linear shapes [cf. A. Kolesárová, Tatra Mt. Math. Publ. 6, 75-82 (1995; Zbl 0851.04005)]. The author indicates that many problems are still open.

MSC:
03E72 Theory of fuzzy sets, etc.
26E50 Fuzzy real analysis
65G30 Interval and finite arithmetic
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