7 research outputs found
Associative polynomial functions over bounded distributive lattices
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.Comment: Final versio
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
On the effect of variable identification on the essential arity of functions
We show that every function of several variables on a finite set of k
elements with n>k essential variables has a variable identification minor with
at least n-k essential variables. This is a generalization of a theorem of
Salomaa on the essential variables of Boolean functions. We also strengthen
Salomaa's theorem by characterizing all the Boolean functions f having a
variable identification minor that has just one essential variable less than f.Comment: 10 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
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
Parametrized arity gap
International audienceWe propose a parametrized version of arity gap. The parametrized arity gap gap (f, ℓ) of a function f:An→B measures the minimum decrease in the number of essential variables of f when ℓ consecutive identifications of pairs of essential variables are performed. We determine gap (f, ℓ) for an arbitrary function f and a nonnegative integer ℓ. We also propose other variants of arity gap and discuss further problems pertaining to the effect of identification of variables on the number of essential variables of functions
A survey on the arity gap
International audienceThe arity gap of a function of several variables is defined as the minimum decrease in the number of essential variables when essential variables of the function are identified. We present a brief survey on the research done on the arity gap, from the first studies of this notion up to recent developments, and discuss some natural extensions and related problems