1,997 research outputs found
Matroidal approaches to rough sets via closure operators
AbstractThis paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems
Rough matroids based on coverings
The introduction of covering-based rough sets has made a substantial
contribution to the classical rough sets. However, many vital problems in rough
sets, including attribution reduction, are NP-hard and therefore the algorithms
for solving them are usually greedy. Matroid, as a generalization of linear
independence in vector spaces, it has a variety of applications in many fields
such as algorithm design and combinatorial optimization. An excellent
introduction to the topic of rough matroids is due to Zhu and Wang. On the
basis of their work, we study the rough matroids based on coverings in this
paper. First, we investigate some properties of the definable sets with respect
to a covering. Specifically, it is interesting that the set of all definable
sets with respect to a covering, equipped with the binary relation of inclusion
, constructs a lattice. Second, we propose the rough matroids based
on coverings, which are a generalization of the rough matroids based on
relations. Finally, some properties of rough matroids based on coverings are
explored. Moreover, an equivalent formulation of rough matroids based on
coverings is presented. These interesting and important results exhibit many
potential connections between rough sets and matroids.Comment: 15page
Continuous selections of multivalued mappings
This survey covers in our opinion the most important results in the theory of
continuous selections of multivalued mappings (approximately) from 2002 through
2012. It extends and continues our previous such survey which appeared in
Recent Progress in General Topology, II, which was published in 2002. In
comparison, our present survey considers more restricted and specific areas of
mathematics. Note that we do not consider the theory of selectors (i.e.
continuous choices of elements from subsets of topological spaces) since this
topics is covered by another survey in this volume
The topological dimension of type I C*-algebras
While there is only one natural dimension concept for separable, metric
spaces, the theory of dimension in noncommutative topology ramifies into
different important concepts. To accommodate this, we introduce the abstract
notion of a noncommutative dimension theory by proposing a natural set of
axioms. These axioms are inspired by properties of commutative dimension
theory, and they are for instance satisfied by the real and stable rank, the
decomposition rank and the nuclear dimension.
We add another theory to this list by showing that the topological dimension,
as introduced by Brown and Pedersen, is a noncommutative dimension theory of
type I C*-algebras. We also give estimates of the real and stable rank of a
type I C*-algebra in terms of its topological dimension.Comment: 20 pages; minor correction
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