Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

Int. J. Mach. Learn. & Cyber.

Wang

2014

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The expansion axiom of matroids requires only the existence of some kind of independent sets, not the uniqueness of them. This causes that the base families of some matroids can be reduced while the unions of the base families of these matroids remain unchanged. In this paper, we define unique expansion matroids in which the expansion axiom has some extent uniqueness; we define union minimal matroids in which the base families have some extent minimality. Some properties of them and the relationship between them are studied. First, we propose the concepts of secondary base and forming base family. Secondly, we propose the concept of unique expansion matroid, and prove that a matroid is a unique expansion matroid if and only if its forming base family is a partition. Thirdly, we propose the concept of union minimal matroid, and prove that unique expansion matroids are union minimal matroids. Finally, we extend the concept of unique expansion matroid to unique exchange matroid and prove that both unique expansion matroids and their dual matroids are unique exchange matroids. (William Zhu) According to the above theorem, we introduce the concept of dual matroid. Definition 7. (Dual matroid [3]) Let M (E, I) be a matroid. (E, Low(Com(B(M )))) is denoted as M * and called the dual matroid of M . Sometimes, we denote I(M * ) and B(M * ) as I * (M ) and B * (M ), respectively. Unique expansion matroidIn this section, we propose the concepts of secondary base, forming base family, unique expansion matroid and unique partition matroid. Then we study their properties and the relationship between them. Secondary base and forming base familySecondary bases are a type of independent sets. We give its definition as follows. Definition 8. (Secondary base) Let M be a matroid and r(M ) > 0. {A ∈ I(M )||A| = r(M ) − 1} is denoted as s(M ) and called the secondary base family of M . Any A ∈ s(M ) is called a secondary base of M .We propose an operator on matroids.It is obvious X ∩ K M (X) = ∅. By the above definition, we give the concept of forming base family. Definition 10. (Forming base family) Let M be a matroid and rGiven a base of a matroid M , we obtain a subset of F (M ). Definition 11. Let M be a matroid, r(M ) > 0 and B ∈ B(M ). The forming base family of M with respect of B is defined by:The forming base family with respect of a base has the following property.For proving the above proposition, we firstly prove the following simple lemma.The proof of Proposition 2 is presented as follows.Proof. If r(M ) = 1, by Lemma 1, this proposition follows. If r(M ) > 1, we letBy Proposition 2, we have the following corollary.Proof. It follows form F M (B) ⊆ F (M ) and Proposition 2.Proposition 9. ( [5,6]) Let E be a finite set and P = {P 1 , P 2 , · · · , P m } be a partition on E. Let k 1 , · · · , k m be a group of nonnegative integers, which satisfyThen (E, I(P ; k 1 , · · · , k m )) is a matroid.By the above proposition, we introduce the following definition.Definition 13. (Partition matroid) Matroid (E, I(P ; k 1 , · · · , k...

“…Matroids have been applied to diverse fields such as algorithm design [1], combinatorial optimization [4]. Recently, matroids have been combined with rough sets [2,6,7,8,10].…”

Section: Introductionmentioning

confidence: 99%

Secondary basis unique augmentation matroids and union minimal matroids

Int. J. Mach. Learn. & Cyber.

Wang

2014

Self Cite

“…As one of the three branches of granular computing, rough set theory is first proposed by Pawlak [20,21] for dealing with vagueness and granularity in information systems. In theory, rough sets have been connected with matroids [10,14,18,27,29,30,37,47], lattices [5,8,16,31], hyperstructure theory [35], topology [12,13,44], fuzzy sets [11,32], and so on. Rough set theory is built on equivalence relations or partitions.…”

Section: Introductionmentioning

confidence: 99%

Applications of repeat degree to coverings of neighborhoods

Int. J. Mach. Learn. & Cyber.

2014

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Covering of neighborhoods is an important concept in covering-based rough sets. There are many unsolved issues related to coverings of neighborhoods. The concept of repeat degree is proposed to study under what condition a covering of neighborhoods is a partition. It enables us to deal with many issues related to coverings of neighborhoods when coverings are incomplete. This paper applies repeat degree to solve some fundamental issues in coverings of neighborhoods. First, we investigate under what condition a covering of neighborhoods is equal to the reduct of the covering which induces the covering of neighborhoods. Then we study under what condition two coverings induce the same relation and the same covering of neighborhoods. Finally, we propose an approach to calculate coverings through repeat degree.

“…Rough set theory, proposed by Pawlak [11,12], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. In theory, rough sets have been connected with matroids [13,16], lattices [3,4,9,15], hyperstructure theory [18], topology [6,7,21], fuzzy sets [5,17], and so on. Rough set theory is built on an equivalence relation, or to say, on a partition.…”

Section: Introductionmentioning

confidence: 99%

Condition for neighbourhoods induced by a covering to be equal to the covering itself

2014

IJGCRSIS

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It is a meaningful issue that under what condition neighborhoods induced by a covering are equal to the covering itself. A necessary and sufficient condition for this issue has been provided by some scholars. In this paper, through a counter-example, we firstly point out the necessary and sufficient condition is false. Second, we present a necessary and sufficient condition for this issue. Third, we concentrate on the inverse issue of computing neighborhoods by a covering, namely giving an arbitrary covering, whether or not there exists another covering such that the neighborhoods induced by it is just the former covering. We present a necessary and sufficient condition for this issue as well. In a word, through the study on the two fundamental issues induced by neighborhoods, we have gained a deeper understanding of the relationship between neighborhoods and the covering which induce the neighborhoods.