arXiv:0808.2619v1 [math.RA] 19 Aug 2008 Representations and characterizations of weighted lattice polynomial functions Miguel Couceiro Mathematics Research Unit, University of Luxembourg 162A, avenue de la Fa¨ ıencerie, L-1511 Luxembourg, Luxembourg miguel.couceiro[at]uni.lu Jean-Luc Marichal Mathematics Research Unit, University of Luxembourg 162A, avenue de la Fa¨ ıencerie, L-1511 Luxembourg, Luxembourg jean-luc.marichal[at]uni.lu August 19, 2008 Abstract Let L be a bounded distributive lattice. In this paper we focus on those functions f : L n → L which can be expressed in the language of bounded lattices using variables and constants, the so-called “weighted” lattice polynomial functions. Clearly, such functions must be nondecreasing in each variable, but the converse does not hold in general. Thus it is natural to ask which nondecreasing functions can be represented by weighted lattice polynomials. We answer this question by providing characteri- zations of weighted lattice polynomial functions given by means of systems of func- tional equations and in terms of necessary and sufficient conditions. We also consider the subclasses of discrete Sugeno integrals, of symmetric functions, and of weighted minimum and maximum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions. Keywords: Distributive lattice; lattice polynomial function; weighted lattice polynomial function; discrete Sugeno integral; normal form; median decomposition; homogeneity; func- tional equation. 1 Introduction This paper deals with functions f : L n → L, where L is a bounded distributive lattice, and which play a fundamental role, not only in universal algebra and lattice theory, but also in computer science, aggregation theory, and decision making. More precisely, we are interested in those functions f : L n → L which can be represented by means of lattice polynomials and, more generally, in those which can be represented by “weighted” lattice polynomials, i.e., expressions written in the language of bounded lattices using both vari- ables and constants. We shall refer to the latter as “weighted lattice polynomial functions” 1