On shape-preserving additions of fuzzy intervals.

*(English)*Zbl 0993.03070For given left and right shapes \(L\) and \(R\), the corresponding \(LR\)-fuzzy intervals are characterized by 4 parameters describing their kernel and their support. Triangular norm-based addition of fuzzy quantities is, in general, a process with high computational complexity. However, in the case of processing \(LR\)-fuzzy numbers (intervals) by means of some specific t-norm preserving the shapes \(L\) and \(R\), the processing is reduced to processing the parameters only, thus efficiently reducing the computational complexity.

Specific cases of shape-preservation were given by A. Kolesárová [Tatra Mt. Math. Publ. 6, 75-81 (1995; Zbl 0851.04005)] (for linear shapes) and by R. Mesiar [Fuzzy Sets Syst. 86, 73-78 (1997; Zbl 0921.04002)], requiring the convexity of composite functions \(fL\) and \(fR\), where \(f\) is an additive generator of the t-norm \(T\) [see E. P. Klement, R. Mesiar and E. Pap, Triangular norms, Dordrecht: Kluwer (2000; Zbl 0972.03002)].

The present paper completely solves the characterization of all shape preserving additions of fuzzy numbers based on continuous t-norms, confirming the hypothesis of Mesiar from 1997. Note that the characterizations of non-continuous t-norms leading to the shape-preserving additions of \(LR\)-fuzzy numbers (intervals) is still an open problems.

Specific cases of shape-preservation were given by A. Kolesárová [Tatra Mt. Math. Publ. 6, 75-81 (1995; Zbl 0851.04005)] (for linear shapes) and by R. Mesiar [Fuzzy Sets Syst. 86, 73-78 (1997; Zbl 0921.04002)], requiring the convexity of composite functions \(fL\) and \(fR\), where \(f\) is an additive generator of the t-norm \(T\) [see E. P. Klement, R. Mesiar and E. Pap, Triangular norms, Dordrecht: Kluwer (2000; Zbl 0972.03002)].

The present paper completely solves the characterization of all shape preserving additions of fuzzy numbers based on continuous t-norms, confirming the hypothesis of Mesiar from 1997. Note that the characterizations of non-continuous t-norms leading to the shape-preserving additions of \(LR\)-fuzzy numbers (intervals) is still an open problems.

Reviewer: Radko Mesiar (Bratislava)

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

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\textit{D. H. Hong}, J. Math. Anal. Appl. 267, No. 1, 369--376 (2002; Zbl 0993.03070)

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