414 research outputs found
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
Function classes and relational constraints stable under compositions with clones
The general Galois theory for functions and relational constraints over
arbitrary sets described in the authors' previous paper is refined by imposing
algebraic conditions on relations
Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices
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 minimum
and maximum functions, and present their characterizations, accordingly.
Moreover, we discuss normal form representations of these functions
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