Aggregation-based extensions of fuzzy measures

We present a method extending fuzzy measures on N={1,…,n} (represented as Boolean utility functions) to n-ary aggregation functions (utility functions) by means of a suitable n-ary aggregation function and the Möbius transform of the considered fuzzy measure. The method generalizes the well-known Lovász and Owen extensions of nondecreasing pseudo-Boolean functions linked to fuzzy measures. All n-ary aggregation functions suitable for the proposed construction are completely characterized, including, among others, all n-ary copulas. Associative extended aggregation functions applicable in the case of an arbitrary arity are also completely characterized.