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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
Rough sets and matroidal contraction
Rough sets are efficient for data pre-processing in data mining. As a
generalization of the linear independence in vector spaces, matroids provide
well-established platforms for greedy algorithms. In this paper, we apply rough
sets to matroids and study the contraction of the dual of the corresponding
matroid. First, for an equivalence relation on a universe, a matroidal
structure of the rough set is established through the lower approximation
operator. Second, the dual of the matroid and its properties such as
independent sets, bases and rank function are investigated. Finally, the
relationships between the contraction of the dual matroid to the complement of
a single point set and the contraction of the dual matroid to the complement of
the equivalence class of this point are studied.Comment: 11 page
Approximations from Anywhere and General Rough Sets
Not all approximations arise from information systems. The problem of fitting
approximations, subjected to some rules (and related data), to information
systems in a rough scheme of things is known as the \emph{inverse problem}. The
inverse problem is more general than the duality (or abstract representation)
problems and was introduced by the present author in her earlier papers. From
the practical perspective, a few (as opposed to one) theoretical frameworks may
be suitable for formulating the problem itself. \emph{Granular operator spaces}
have been recently introduced and investigated by the present author in her
recent work in the context of antichain based and dialectical semantics for
general rough sets. The nature of the inverse problem is examined from
number-theoretic and combinatorial perspectives in a higher order variant of
granular operator spaces and some necessary conditions are proved. The results
and the novel approach would be useful in a number of unsupervised and semi
supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings,
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