# Quasi-polynomial functions over bounded distributive lattices - Mathematics > Functional Analysis

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Abstract: In arXiv 0811.3913 the authors introduced the notion of quasi-polynomialfunction as being a mapping f: X^n -> X defined and valued on a bounded chain Xand which can be factorized as fx 1,

.,x n=pphix 1,

.,phix n, where pis a polynomial function i.e., a combination of variables and constants usingthe chain operations - and and phi is an order-preserving map. In the currentpaper we study this notion in the more general setting where the underlyingdomain and codomain sets are, possibly different, bounded distributivelattices, and where the inner function is not necessarily order-preserving.These functions appear naturally within the scope of decision making underuncertainty since, as shown in this paper, they subsume overall preferencefunctionals associated with Sugeno integrals whose variables are transformed bya given utility function. To axiomatize the class of quasi-polynomialfunctions, we propose several generalizations of well-established properties inaggregation theory, as well as show that some of the characterizations given inarXiv 0811.3913 still hold in this general setting. Moreover, we investigatethe so-called transformed polynomial functions essentially, compositions ofunary mappings with polynomial functions and show that, under certainconditions, they reduce to quasi-polynomial functions.

Author: ** Miguel Couceiro, Jean-Luc Marichal**

Source: https://arxiv.org/