A matroidal approach to rough set theory

Tang, Jianguo; She, Kun; Min, Fan; Zhu, William

doi:10.1016/j.tcs.2012.10.060

Cited by 42 publications

(12 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Granular Computing (briefly GrC) is an emerging paradigm which relies on the idea of partitioning a set of objects in some granules depending on some given criteria [29,30,38,39]. Many ideas and methods of GrC have been used in order to investigate discrete mathematical objects, such as matroids, set partitions and ordered structures [21,25,26,36,37].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lattice Representations with Set Partitions Induced by Pairings

Chiaselotti¹,

Gentile²,

Infusino³

2020

Electron. J. Combin.

View full text Add to dashboard Cite

We call a quadruple $\mathcal{W}:=\langle F,U,\Omega,\Lambda \rangle$, where $U$ and $\Omega$ are two given non-empty finite sets, $\Lambda$ is a non-empty set and $F$ is a map having domain $U\times \Omega$ and codomain $\Lambda$, a pairing on $\Omega$. With this structure we associate a set operator $M_{\mathcal{W}}$ by means of which it is possible to define a preorder $\ge_{\mathcal{W}}$ on the power set $\mathcal{P}(\Omega)$ preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice $\mathbb{L}$ there exist a finite set $\Omega_{\mathbb{L}}$ and a pairing $\mathcal{W}$ on $\Omega_\mathbb{L}$ such that the quotient of the preordered set $(\mathcal{P}(\Omega_\mathbb{L}), \ge_\mathcal{W})$ with respect to its symmetrization is a lattice that is order-isomorphic to $\mathbb{L}$. In the second result, we prove that when the lattice $\mathbb{L}$ is endowed with an order-reversing involutory map $\psi: L \to L$ such that $\psi(\hat 0_{\mathbb{L}})=\hat 1_{\mathbb{L}}$, $\psi(\hat 1_{\mathbb{L}})=\hat 0_{\mathbb{L}}$, $\psi(\alpha) \wedge \alpha=\hat 0_{\mathbb{L}}$ and $\psi(\alpha) \vee \alpha=\hat 1_{\mathbb{L}}$, there exist a finite set $\Omega_{\mathbb{L},\psi}$ and a pairing on it inducing a specific poset which is order-isomorphic to $\mathbb{L}$.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Based on recent results obtained in the graph context [13,14] and on other general results linked to theoretical computer science [1,2,12,15,25,26,36,37,39], in this paper we continue the study of the basic property of the set system M(W) and use methods from GrC in relation to order-theory, limited to finite lattices.…”

Section: Introductionmentioning

confidence: 99%

Lattice Representations with Set Partitions Induced by Pairings

Chiaselotti¹,

Gentile²,

Infusino³

2020

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…Recently, the interrelations between rough set theory and various branches of mathematics, such as combinatorics [12], monoids [10], matroids [15,23,24,25], groups [7,18], integral domains [11] and modules [9] has been deeply studied and constitute a research field which is developing rapidly. In our perspective, we are interested in the interrelations between rough set theory and groups.…”

Section: Introductionmentioning

confidence: 99%

Rough action on topological rough groups

et al. 2020

View full text Add to dashboard Cite

<p>In this paper we explore the interrelations between rough set theory and group theory. To this end, we first define a topological rough group homomorphism and its kernel. Moreover, we introduce rough action and topological rough group homeomorphisms, providing several examples. Next, we combine these two notions in order to define topological rough homogeneous spaces, discussing results concerning open subsets in topological rough groups.</p>

show abstract

“…Studying rough sets with matroids is helpful to enrich the theory system and to extend the applications of rough sets. Some works on the connection between rough sets and matroids have been conducted [6,8,11,21,20,29,28,26,39,40].…”

Section: Introductionmentioning

confidence: 99%

Geometric Lattice Structure of Covering and Its Application to Attribute Reduction through Matroids

Huang

Zhu

2014

Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice to study the attribute reduction issues of information systems.

show abstract

A matroidal approach to rough set theory

Cited by 42 publications

References 34 publications

Lattice Representations with Set Partitions Induced by Pairings

Lattice Representations with Set Partitions Induced by Pairings

Rough action on topological rough groups

Geometric Lattice Structure of Covering and Its Application to Attribute Reduction through Matroids

Contact Info

Product

Resources

About