On the existence of aggregation functions with given super-additive and sub-additive transformations

Šipošová, Alexandra; Šipeky, Ladislav and Širáň, Jozef (2017). On the existence of aggregation functions with given super-additive and sub-additive transformations. Fuzzy Sets and Systems, 324 pp. 117–126.

DOI: https://doi.org/10.1016/j.fss.2016.11.009

Abstract

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, A → A^∗ and A → A^∗, of an aggregation function A. We prove that if A^∗ has a slightly stronger property of being strictly directionally convex, then A = A^∗ and A_∗ is linear; dually, if A_∗ is strictly directionally concave, then A = A_∗ and A^∗ is linear. This implies, for example, the existence of pairs of functions f≤g sub-additive and super-additive on [0, ∞[ⁿ, respectively, with zero value at the origin and satisfying relatively mild extra conditions, for which there exists no aggregation function A on [0, ∞[ⁿ such that A^∗=f and A^∗=g.

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