Topological structures of a type of granule based covering rough sets

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3129-3141
Author(s):  
Yan-Lan Zhang ◽  
Chang-Qing Li
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Topological Properties ◽  
Binary Relations ◽  
Upper Approximation ◽  
Approximation Operators

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindel?f property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

scholarly journals Topological properties of a pair of relation-based approximation operators

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6175-6183
Author(s):  
Yan-Lan Zhang ◽  
Chang-Qing Li
Keyword(s):  
Data Mining ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Equivalence Relations ◽  
Topological Properties ◽  
Upper Approximation ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Basic Concepts

Rough set theory is an important tool for data mining. Lower and upper approximation operators are two important basic concepts in the rough set theory. The classical Pawlak rough approximation operators are based on equivalence relations and have been extended to relation-based generalized rough approximation operators. This paper presents topological properties of a pair of relation-based generalized rough approximation operators. A topology is induced by the pair of generalized rough approximation operators from an inverse serial relation. Then, connectedness, countability, separation property and Lindel?f property of the topological space are discussed. The results are not only beneficial to obtain more properties of the pair of approximation operators, but also have theoretical and actual significance to general topology.

Approximations in Rough Sets vs Granular Computing for Coverings

2010 ◽  
Vol 4 (2) ◽  
pp. 63-76 ◽  
Author(s):  
Guilong Liu ◽  
William Zhu
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Knowledge Discovery In Databases ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Covering Rough Set

Rough set theory is an important technique in knowledge discovery in databases. Classical rough set theory proposed by Pawlak is based on equivalence relations, but many interesting and meaningful extensions have been made based on binary relations and coverings, respectively. This paper makes a comparison between covering rough sets and rough sets based on binary relations. This paper also focuses on the authors’ study of the condition under which the covering rough set can be generated by a binary relation and the binary relation based rough set can be generated by a covering.

On Characterization of Relation Based Rough Set Algebras

2010 ◽  
pp. 69-91
Author(s):  
Wei-Zhi Wu ◽  
Wen-Xiu Zhang
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Axiomatic Approach ◽  
Topological Spaces ◽  
Upper Approximation ◽  
Fuzzy Approximation ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Fuzzy Topological Spaces

Rough set theory is one of the most advanced areas popularizing GrC. The basic notions in rough set theory are the lower and upper approximation operators. A rough set algebra is a set algebra with two additional lower and upper approximation operators. In this chapter, we analyze relation based rough set algebras in both crisp and fuzzy environments. We first review the constructive definitions of generalized crisp rough approximation operators, rough fuzzy approximation operators, and fuzzy rough approximation operators. We then present the essential properties of the corresponding lower and upper approximation operators. We also characterize the approximation operators by using the axiomatic approach. Finally, the connection between fuzzy rough set algebras and fuzzy topological spaces is established.

Some Remarks on the Concept of Approximations from the View of Knowledge Engineering

2010 ◽  
Vol 4 (2) ◽  
pp. 1-11 ◽  
Author(s):  
Tsau Young Lin ◽  
Rushin Barot ◽  
Shusaku Tsumoto
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Knowledge Engineering ◽  
Topological Spaces

The concepts of approximations in granular computing (GrC) vs. rough set theory (RS) are examined. Examples are constructed to contrast their differences in the Global GrC Model (2nd GrC Model), which, in pre-GrC term, is called partial coverings. Mathematically speaking, RS-approximations are “sub-base” based, while GrC-approximations are “base” based, where “sub-base” and “base” are two concepts in topological spaces. From the view of knowledge engineering, its meaning in RS-approximations is rather obscure, while in GrC, it is the concept of knowledge approximations.

Approximations in Rough Sets vs Granular Computing for Coverings

2012 ◽  
pp. 152-163
Author(s):  
Guilong Liu ◽  
William Zhu
Keyword(s):  
Knowledge Discovery ◽  
Binary Relation ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Knowledge Discovery In Databases ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Covering Rough Set

Rough set theory is an important technique in knowledge discovery in databases. Classical rough set theory proposed by Pawlak is based on equivalence relations, but many interesting and meaningful extensions have been made based on binary relations and coverings, respectively. This paper makes a comparison between covering rough sets and rough sets based on binary relations. This paper also focuses on the authors’ study of the condition under which the covering rough set can be generated by a binary relation and the binary relation based rough set can be generated by a covering.

scholarly journals Multigranulations Rough Set Method of Attribute Reduction in Information Systems Based on Evidence Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Minlun Yan
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Decision Rules ◽  
Evidence Theory ◽  
Point Of View ◽  
Upper Approximation

Attribute reduction is one of the most important problems in rough set theory. However, from the granular computing point of view, the classical rough set theory is based on a single granulation. It is necessary to study the issue of attribute reduction based on multigranulations rough set. To acquire brief decision rules from information systems, this paper firstly investigates attribute reductions by combining the multigranulations rough set together with evidence theory. Concepts of belief and plausibility consistent set are proposed, and some important properties are addressed by the view of the optimistic and pessimistic multigranulations rough set. What is more, the multigranulations method of the belief and plausibility reductions is constructed in the paper. It is proved that a set is an optimistic (pessimistic) belief reduction if and only if it is an optimistic (pessimistic) lower approximation reduction, and a set is an optimistic (pessimistic) plausibility reduction if and only if it is an optimistic (pessimistic) upper approximation reduction.

scholarly journals Numerical Characterizations of Topological Reductions of Covering Information Systems in Evidence Theory

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yan-Lan Zhang ◽  
Chang-Qing Li
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Evidence Theory ◽  
Topological Spaces ◽  
Relative Significance ◽  
Approximation Operators ◽  
Covering Rough Set

The reductions of covering information systems in terms of covering approximation operators are one of the most important applications of covering rough set theory. Based on the connections between the theory of topology and the covering rough set theory, two kinds of topological reductions of covering information systems are discussed in this paper, which are characterized by the belief and plausibility functions from the evidence theory. The topological spaces by two pairs of covering approximation operators in covering information systems are pseudo-discrete, which deduce partitions. Then, using plausibility function values of the sets in the partitions, the definitions of significance and relative significance of coverings are presented. Hence, topological reduction algorithms based on the evidence theory are proposed in covering information systems, and an example is adopted to illustrate the validity of the algorithms.

Some Remarks on the Concept of Approximations from the View of Knowledge Engineering

2012 ◽  
pp. 278-286
Author(s):  
Tsau Young Lin ◽  
Rushin Barot ◽  
Shusaku Tsumoto
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Knowledge Engineering ◽  
Topological Spaces

The concepts of approximations in granular computing (GrC) vs. rough set theory (RS) are examined. Examples are constructed to contrast their differences in the Global GrC Model (2nd GrC Model), which, in pre-GrC term, is called partial coverings. Mathematically speaking, RS-approximations are “sub-base” based, while GrC-approximations are “base” based, where “sub-base” and “base” are two concepts in topological spaces. From the view of knowledge engineering, its meaning in RS-approximations is rather obscure, while in GrC, it is the concept of knowledge approximations.

scholarly journals Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations

2022 ◽  
Vol 7 (2) ◽  
pp. 2891-2928
Author(s):  
Rukchart Prasertpong ◽  
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Equivalence Relations ◽  
Prime Ideals ◽  
Fuzzy Relations ◽  
Binary Relations ◽  
Soft Sets ◽  
Rough Fuzzy Set

<abstract><p>In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.</p></abstract>

THE INFORMATION ENTROPY, ROUGH ENTROPY AND KNOWLEDGE GRANULATION IN ROUGH SET THEORY

2004 ◽  
Vol 12 (01) ◽  
pp. 37-46 ◽  
Author(s):  
JIYE LIANG ◽  
ZHONGZHI SHI
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Information Entropy ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Computer Applications ◽  
Mathematical Tool

Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances which are characterized by vagueness and uncertainty. In this paper, we introduce the concepts of information entropy, rough entropy and knowledge granulation in rough set theory, and establish the relationships among those concepts. These results will be very helpful for understanding the essence of concept approximation and establishing granular computing in rough set theory.

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