Andrea Stupňanová

Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on ${[0,1]}^n$, namely, the directional shift stability. This new property extends the standard shift invariantness (difference scale invariantness), which can be seen as a particular directional shift stability. The newly proposed property can also be seen as a particular kind of local linearity. Several examples and a complete characterization for the case of $n=2$ of directionally shift-stable aggregation and pre-aggregation functions are also given.

In this work we propose a generalization of the Choquet integral starting from its definition in terms of the Mobius transform. We modify the product on the set of real numbers considered in the Lovasz 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, independently of the considered fuzzy measure, is given, and several particular examples are presented. Finally, all functions $F$ yielding an aggregation function, independently 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.