Quasi-Polynomial Functions on Bounded Chains

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 present complete descriptions of 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 quasipolynomial function which appears as a natural extension of the well-established notion of polynomial function. We present 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 present characterizations accordingly. Keywords— Discrete Sugeno integral, quasi-polynomial function, horizontal maxitivity and minitivity, comonotonic maxitivity and minitivity, functional equation.