We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions.

Pippenger’s Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set A and taking values in a possibly different set B, where any or both of A and B may be finite or infinite.

Training set extension is an important issue in machine learning. Indeed when the examples at hand are in a limited quantity, the performances of standard classifiers may significantly decrease and it can be helpful to build additional examples. In this paper, we consider the use of analogical reasoning, and more particularly of analogical proportions for extending training sets. Here the ground truth labels are considered to be given by a (partially known) function. We examine the conditions that are required for such functions to ensure an error-free extension in a Boolean setting. To this end, we introduce the notion of Analogy Preserving (AP) functions, and we prove that their class is the class of affine Boolean functions. This noteworthy theoretical result is complemented with an empirical investigation of approximate AP functions, which suggests that they remain suitable for training set extension.

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n ⩾ 1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same.