The arity gap of order-preserving functions and extensions of pseudo-Boolean functions

The aim of this paper is to classify order-preserving functions according to their arity gap. Noteworthy examples of order-preserving functions are so-called aggregation functions. We first explicitly classify the Lov\'asz extensions of pseudo-Boolean functions according to their arity gap. Then we consider the class of order-preserving functions between partially ordered sets, and establish a similar explicit classification for this function class.Comment: 11 pages, material reorganize

Generalizations of Swierczkowski's lemma and the arity gap of finite functions

Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an at least quaternary operation on a finite set A and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of Swierczkowski's Lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A.Comment: 11 pages, proofs simplified, contents reorganize

The arity gap of polynomial functions over bounded distributive lattices

Let A and B be arbitrary sets with at least two elements. The arity gap of a function f: A^n \to B is the minimum decrease in its essential arity when essential arguments of f are identified. In this paper we study the arity gap of polynomial functions over bounded distributive lattices and present a complete classification of such functions in terms of their arity gap. To this extent, we present a characterization of the essential arguments of polynomial functions, which we then use to show that almost all lattice polynomial functions have arity gap 1, with the exception of truncated median functions, whose arity gap is 2.Comment: 7 page

A generalization of Goodstein's theorem: interpolation by polynomial functions of distributive lattices

We consider the problem of interpolating functions partially defined over a distributive lattice, by means of lattice polynomial functions. Goodstein's theorem solves a particular instance of this interpolation problem on a distributive lattice L with least and greatest elements 0 and 1, resp.: Given an n-ary partial function f over L, defined on all 0-1 tuples, f can be extended to a lattice polynomial function p over L if and only if f is monotone; in this case, the interpolating polynomial p is unique. We extend Goodstein's theorem to a wider class of n-ary partial functions f over a distributive lattice L, not necessarily bounded, where the domain of f is a cuboid of the form D={a1,b1}x...x{an,bn} with ai<bi, and determine the class of such partial functions which can be interpolated by lattice polynomial functions. In this wider setting, interpolating polynomials are not necessarily unique; we provide explicit descriptions of all possible lattice polynomial functions which interpolate these partial functions, when such an interpolation is available.Comment: 12 page

Galois connection for sets of operations closed under permutation, cylindrification and composition

We consider sets of operations on a set A that are closed under permutation of variables, addition of dummy variables and composition. We describe these closed sets in terms of a Galois connection between operations and systems of pointed multisets, and we also describe the closed sets of the dual objects by means of necessary and sufficient closure conditions. Moreover, we show that the corresponding closure systems are uncountable for every A with at least two elements.Comment: 22 pages; Section 4 adde

Pseudo-polynomial functions over finite distributive lattices

In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice polynomial function over Y, and each uk is a map from Xk to Y. The resulting functions are referred to as pseudo-polynomial functions. We present an axiomatization for this class of pseudo-polynomial functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-polynomial function.Comment: 16 pages, 2 figure