627 research outputs found
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
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
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
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