  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 Lov\'asz 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 Lov\'asz extensions, which includes the discrete symmetric
Choquet integrals.
