Axiomatizations of quasi-polynomial functions on bounded chains

TL;DR

This paper characterizes classes of functions on bounded chains using properties like maxitivity and introduces quasi-polynomial functions as an extension of polynomial functions, providing axiomatizations and exploring subclasses.

Contribution

It introduces and axiomatizes quasi-polynomial functions on bounded chains, extending polynomial functions and analyzing their properties and subclasses.

Findings

Complete axiomatizations of functions with horizontal and comonotonic maxitivity.

Introduction of quasi-polynomial functions as a natural extension of polynomial functions.

Characterizations of subclasses like quasi-term and quasi-weighted maximum/minimum functions.

Abstract

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.