Abstract
Four different types of convolutions of aggregation functions (the upper, the lower, the super-, and the sub-convolution) are examined in the setting of both sub- and super-decomposition integrals defined on a finite space. Examples of the results of the paper are provided. As a by-product, the super-additive transformation of sub-decomposition integrals and the sub-additive transformation of super-decomposition integrals are fully characterized. Possible applications are indicated.
Highlights
One may notice that the concept of aggregation functions plays an important role in the decision theory, and the concept of convolution is important in the classical analysis, and in probability, acoustics, image processing, computer vision, etc
We solved the problem of the upper convolution and super convolution of sub-decomposition integrals with respect to the same monotone measure and, analogously, the lower convolution and sub-convolution of super-decomposition integrals with respect to the same monotone measure
But replacing the sub-collection integrals with sub-decomposition integrals? Or, is it possible to characterize those decomposition systems for which the lower convolution of sub-decomposition integrals is again a sub-decomposition integral? Another interesting question is related to the fact that some decomposition integrals are extensions of the Lebesgue integral; for more details, see [14]
Summary
Academic Editors: Anca Croitoru, Radko Mesiar, Anna Rita Sambucini and Bianca Satco. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Inspired by convolutions of aggregation functions proposed in [1], in this paper, we apply these convolutions in the setting of sub-decomposition integrals [2] and super-decomposition integrals [3], a special class of aggregation functions that includes many well-known non-linear integrals, such as the Choquet integral [4], the Shilkret integral [5], the PAN integral [6,7], the concave integral [8], or the convex integral [3]. Note that these integrals contribute to the basics of set-valued analysis, see e.g., [9,10].
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