Möbius product-based constructions of aggregation functions

Möbius transforms of capacities or games were considered as a tool for extensions of capacities to particular aggregation functions in several papers. This is, for example, the case of the Lovász extension coinciding with the Choquet integral, or the Owen extension that is also known as multilinear extension. In this paper, a much deeper study of the links between the Möbius transforms of real-valued functions defined on finite bounded posets and aggregation functions is performed. We introduce a Möbius product of any two real-valued functions defined on an arbitrary finite bounded poset, and then, fixing the poset , , we propose and discuss a construction method for -ary aggregation functions based on the Möbius product of any capacity on and a real-valued function , , defined on and determined by some appropriate -ary aggregation function. We provide some necessary and some sufficient conditions for the introduced construction to yield an aggregation function for any capacity. For the binary case, a complete characterization of conditions under which our approach results in an aggregation function for each capacity is given.