[en] aggregation function ; discrete Choquet integral ; discrete symmetric Choquet integral ; Lovász extension ; functional equation ; Cauchy equation ; comonotonic additivity ; horizontal additivity ; axiomatization
[en] Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals.
[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
