Axiomatizations of quasi-polynomial functions on bounded chains

English

Couceiro, Miguel[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit > ; University Paris-Dauphine > Lamsade]

Marichal, Jean-Luc[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]

[en] aggregation function ; discrete Sugeno integral ; polynomial function ; quasi-polynomial function ; horizontal maxitivity and minitivity ; comonotonic maxitivity and minitivity ; functional equation

[en] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely describe the function classes axiomatized by each of these properties, up to weak versions of monotonicity in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We give further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and provide characterizations accordingly.