Representations and characterizations of weighted lattice polynomial functions

Let L be a bounded distributive lattice. In this paper we focus on those functions f : L → 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 characterizations of weighted lattice polynomial functions given by means of systems of functional 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.