Reference : Quasi-polynomial functions over bounded distributive lattices
 Document type : Scientific journals : Article Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/2858
 Title : Quasi-polynomial functions over bounded distributive lattices Language : English Author, co-author : Couceiro, Miguel [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Marichal, Jean-Luc [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Publication date : Dec-2010 Journal title : Aequationes Mathematicae Publisher : Springer Volume : 80 Issue/season : 3 Pages : 319-334 Peer reviewed : Yes (verified by ORBilu) Audience : International ISSN : 0001-9054 e-ISSN : 1420-8903 City : Basel Country : Switzerland Keywords : [en] Distributive lattice ; Polynomial function ; Quasi-polynomial function ; Functional equation ; Aggregation function ; Discrete Sugeno integral ; Utility function Abstract : [en] In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))$, where $p$ is a polynomial function (i.e., a combination of variables and constants using the chain operations $\wedge$ and $\vee$) and $\varphi$ 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 [6] 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. Permalink : http://hdl.handle.net/10993/2858 DOI : 10.1007/s00010-010-0039-9 Other URL : http://arxiv.org/abs/0909.3009

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