A Generalization of the Choquet Integral Defined in Terms of the Möbius Transform

In this article, we propose a generalization of the Choquet integral, starting from its definition in terms of the Möbius transform. We modify the product on R considered in the Lovász extension form of the Choquet integral into a function F, and we discuss the properties of this new functional. For a fixed n, a complete description of all F yielding an n-ary aggregation function with a fixed diagonal section, independent of the considered fuzzy measure, is given, and several particular examples are presented. Finally, all functions F yielding an aggregation function, independent of the number n of inputs and of the considered fuzzy measure, are characterized, and related aggregation functions are shown to be just the Choquet integrals over the distorted inputs.