# Axiomatizations of Lovász extensions of pseudo-Boolean functions - Mathematics > Functional Analysis

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Abstract: Three important properties in aggregation theory are investigated, namelyhorizontal min-additivity, horizontal max-additivity, and comonotonicadditivity, which are defined by certain relaxations of the Cauchy functionalequation in several variables. We show that these properties are equivalent andwe completely describe the functions characterized by them. By adding someregularity conditions, these functions coincide with the Lov\-asz extensionsvanishing at the origin, which subsume the discrete Choquet integrals. We alsopropose a simultaneous generalization of horizontal min-additivity andhorizontal max-additivity, called horizontal median-additivity, and we describethe corresponding function class. Additional conditions then reduce this classto that of symmetric Lov\-asz extensions, which includes the discrete symmetricChoquet integrals.

Author: ** Miguel Couceiro, Jean-Luc Marichal**

Source: https://arxiv.org/