22,172 research outputs found
Characteristic of partition-circuit matroid through approximation number
Rough set theory is a useful tool to deal with uncertain, granular and
incomplete knowledge in information systems. And it is based on equivalence
relations or partitions. Matroid theory is a structure that generalizes linear
independence in vector spaces, and has a variety of applications in many
fields. In this paper, we propose a new type of matroids, namely,
partition-circuit matroids, which are induced by partitions. Firstly, a
partition satisfies circuit axioms in matroid theory, then it can induce a
matroid which is called a partition-circuit matroid. A partition and an
equivalence relation on the same universe are one-to-one corresponding, then
some characteristics of partition-circuit matroids are studied through rough
sets. Secondly, similar to the upper approximation number which is proposed by
Wang and Zhu, we define the lower approximation number. Some characteristics of
partition-circuit matroids and the dual matroids of them are investigated
through the lower approximation number and the upper approximation number.Comment: 12 page
Parametric matroid of rough set
Rough set is mainly concerned with the approximations of objects through an
equivalence relation on a universe. Matroid is a combinatorial generalization
of linear independence in vector spaces. In this paper, we define a parametric
set family, with any subset of a universe as its parameter, to connect rough
sets and matroids. On the one hand, for a universe and an equivalence relation
on the universe, a parametric set family is defined through the lower
approximation operator. This parametric set family is proved to satisfy the
independent set axiom of matroids, therefore it can generate a matroid, called
a parametric matroid of the rough set. Three equivalent representations of the
parametric set family are obtained. Moreover, the parametric matroid of the
rough set is proved to be the direct sum of a partition-circuit matroid and a
free matroid. On the other hand, since partition-circuit matroids were well
studied through the lower approximation number, we use it to investigate the
parametric matroid of the rough set. Several characteristics of the parametric
matroid of the rough set, such as independent sets, bases, circuits, the rank
function and the closure operator, are expressed by the lower approximation
number.Comment: 15 page
Some characteristics of matroids through rough sets
At present, practical application and theoretical discussion of rough sets
are two hot problems in computer science. The core concepts of rough set theory
are upper and lower approximation operators based on equivalence relations.
Matroid, as a branch of mathematics, is a structure that generalizes linear
independence in vector spaces. Further, matroid theory borrows extensively from
the terminology of linear algebra and graph theory. We can combine rough set
theory with matroid theory through using rough sets to study some
characteristics of matroids. In this paper, we apply rough sets to matroids
through defining a family of sets which are constructed from the upper
approximation operator with respect to an equivalence relation. First, we prove
the family of sets satisfies the support set axioms of matroids, and then we
obtain a matroid. We say the matroids induced by the equivalence relation and a
type of matroid, namely support matroid, is induced. Second, through rough
sets, some characteristics of matroids such as independent sets, support sets,
bases, hyperplanes and closed sets are investigated.Comment: 13 page
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