655 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
Invariant functionals on completely distributive lattices
In this paper we are interested in functionals defined on completely
distributive lattices and which are invariant under mappings preserving
{arbitrary} joins and meets. We prove that the class of nondecreasing invariant
functionals coincides with the class of Sugeno integrals associated with
-valued capacities, the so-called term functionals, thus extending
previous results both to the infinitary case as well as to the realm of
completely distributive lattices. Furthermore, we show that, in the case of
functionals over complete chains, the nondecreasing condition is redundant.
Characterizations of the class of Sugeno integrals, as well as its superclass
comprising all polynomial functionals, are provided by showing that the
axiomatizations (given in terms of homogeneity) of their restriction to
finitary functionals still hold over completely distributive lattices. We also
present canonical normal form representations of polynomial functionals on
completely distributive lattices, which appear as the natural extensions to
their finitary counterparts, and as a by-product we obtain an axiomatization of
complete distributivity in the case of bounded lattices
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
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Pivotal decompositions of functions
We extend the well-known Shannon decomposition of Boolean functions to more
general classes of functions. Such decompositions, which we call pivotal
decompositions, express the fact that every unary section of a function only
depends upon its values at two given elements. Pivotal decompositions appear to
hold for various function classes, such as the class of lattice polynomial
functions or the class of multilinear polynomial functions. We also define
function classes characterized by pivotal decompositions and function classes
characterized by their unary members and investigate links between these two
concepts
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