Quasi-polynomial functions over bounded distributive lattices

TL;DR

This paper extends the concept of quasi-polynomial functions from chains to bounded distributive lattices, exploring their properties, axiomatizations, and applications in decision making under uncertainty.

Contribution

It generalizes quasi-polynomial functions to bounded distributive lattices, introduces new axiomatizations, and connects them to preference functionals and Sugeno integrals.

Findings

Quasi-polynomial functions are characterized in the more general lattice setting.

Some transformed polynomial functions reduce to quasi-polynomial functions under certain conditions.

The paper links these functions to decision making models involving Sugeno integrals.

Abstract

In [arXiv 0811.3913] the authors introduced the notion of quasi-polynomial function as being a mapping f: X^n -> X defined and valued on a bounded chain X and which can be factorized as f(x_1,...,x_n)=p(phi(x_1),...,phi(x_n)), where p is a polynomial function (i.e., a combination of variables and constants using the chain operations / and) and phi is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [arXiv 0811.3913] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions.