## On the convergence of $$T$$-sum of $$L$$-$$R$$ fuzzy numbers.(English)Zbl 0844.04004

Summary: This paper presents the membership function of infinite (or finite) sum (defined by the sup-$$t$$-norm convolution) of $$L$$-$$R$$ fuzzy numbers under the conditions of the convexity of additive generators and the concavity of $$L$$, $$R$$. As an application, we shall calculate the membership function of the limit distribution of the Hamacher sum ($$H_r$$-sum) for $$0\leq r\leq 2$$, which generalizes R. Fullér’s results [ibid. 41, 83-87 (1991; Zbl 0725.04002) and ibid. 42, 205-212 (1991; Zbl 0734.04004)] in the case $$r\in \{0, 1, 2\}$$.

### MSC:

 03E72 Theory of fuzzy sets, etc. 40A99 Convergence and divergence of infinite limiting processes 26E50 Fuzzy real analysis

### Citations:

Zbl 0725.04002; Zbl 0734.04004
Full Text:

### References:

 [1] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Automat. Control, 26, 926-936 (1981) · Zbl 1457.68262 [2] Fullér, R., On product-sum of triangular fuzzy numbers, Fuzzy Sets and Systems, 41, 83-87 (1991) · Zbl 0725.04002 [3] Fullér, R., On Hamacher-sum of triangular fuzzy numbers, Fuzzy Sets and Systems, 42, 205-212 (1991) · Zbl 0734.04004 [4] Fullér, R.; Keresztfalvi, T., t-Norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51, 155-159 (1992) [6] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001 [7] Triesch, E., On the convergence of product-sum series of $$L-R$$ fuzzy numbers, Fuzzy Sets and Systems, 53, 189-192 (1993) · Zbl 0874.26019
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.