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We introduce the concept of quasi-Lovasz extension as being a mapping $${f\colon I^n\to\mathbb{R}}$$ defined on a nonempty real interval I containing the origin and which can be factorized as f(x 1, . . . , x n ) = L(φ(x 1), . . . , φ(x n )), where L is the Lovasz extension of a pseudo-Boolean function $${\psi\colon \{0, 1\}^n \to \mathbb{R}}$$ (i.e., the function $${L\colon \mathbb{R}^n \to \mathbb{R}}$$ whose restriction to each simplex of the standard triangulation of [0, 1] n is the unique affine function which agrees with ψ at the vertices of this simplex) and $${\varphi\colon I \to \mathbb{R}}$$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lovasz extensions, we propose generalizations of properties used to characterize Lovasz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lovasz extensions, which are compositions of symmetric Lovasz extensions with 1-place nondecreasing odd functions.

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.