scholarly journals Topological Interpretation of Rough Sets

2014 ◽  
Vol 22 (1) ◽  
pp. 89-97 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Rough Sets ◽  
Upper Bounds ◽  
Topological Spaces ◽  
Approximation Spaces ◽  
Extremally Disconnected ◽  
Topological Interpretation ◽  
Topological Closure ◽  
Mizar Mathematical Library ◽  
Abstract Spaces

Summary Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi [13] (formalized in Mizar in [14]) and about fourteen sets of Kuratowski [17] (encoded with the help of Mizar adjectives and clusters’ registrations in [1]) in terms of rough approximations. The upper bounds which were 14 and 7 in the original paper of Kuratowski, in our case are six and three, respectively. It turns out that within the classification given by Isomichi, 1st class subsets are precisely crisp sets, 2nd class subsets are proper rough sets, and there are no 3rd class subsets in topological spaces generated by approximations. Also the important results about these spaces is that they are extremally disconnected [15], hence lattices of their domains are Boolean. Furthermore, we develop the theory of abstract spaces equipped with maps possessing characteristic properties of rough approximations which enables us to freely use the notions from the theory of rough sets and topological spaces formalized in the Mizar Mathematical Library [10].

scholarly journals A New Single-Valued Neutrosophic Rough Sets and Related Topology

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Qiu Jin ◽  
Kai Hu ◽  
Chunxin Bo ◽  
Lingqiang Li
Keyword(s):  
Rough Sets ◽  
Topological Spaces ◽  
Approximation Operator ◽  
Approximation Space ◽  
Fuzzy Rough Sets ◽  
Approximation Spaces ◽  
Lower Approximation ◽  
Basic Properties ◽  
Approximation Operators ◽  
New Type

(Fuzzy) rough sets are closely related to (fuzzy) topologies. Neutrosophic rough sets and neutrosophic topologies are extensions of (fuzzy) rough sets and (fuzzy) topologies, respectively. In this paper, a new type of neutrosophic rough sets is presented, and the basic properties and the relationships to neutrosophic topology are discussed. The main results include the following: (1) For a single-valued neutrosophic approximation space U , R , a pair of approximation operators called the upper and lower ordinary single-valued neutrosophic approximation operators are defined and their properties are discussed. Then the further properties of the proposed approximation operators corresponding to reflexive (transitive) single-valued neutrosophic approximation space are explored. (2) It is verified that the single-valued neutrosophic approximation spaces and the ordinary single-valued neutrosophic topological spaces can be interrelated to each other through our defined lower approximation operator. Particularly, there is a one-to-one correspondence between reflexive, transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces.

scholarly journals Binary Relations-based Rough Sets – an Automated Approach

2016 ◽  
Vol 24 (2) ◽  
pp. 143-155 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Binary Relation ◽  
Rough Sets ◽  
The State ◽  
General Mechanism ◽  
Binary Relations ◽  
Approximation Spaces ◽  
Approximation Operators ◽  
Mizar Mathematical Library ◽  
Almost All

Summary Rough sets, developed by Zdzisław Pawlak [12], are an important tool to describe the state of incomplete or partially unknown information. In this article, which is essentially the continuation of [8], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library [11]). Here we drop the classical equivalence- and tolerance-based models of rough sets trying to formalize some parts of [18]. The main aim of this Mizar article is to provide a formal counterpart for the rest of the paper of William Zhu [18]. In order to do this, we recall also Theorem 3 from Y.Y. Yao’s paper [17]. The first part of our formalization (covering first seven pages) is contained in [8]. Now we start from page 5003, sec. 3.4. [18]. We formalized almost all numbered items (definitions, propositions, theorems, and corollaries), with the exception of Proposition 7, where we stated our theorem only in terms of singletons. We provided more thorough discussion of the property positive alliance and its connection with seriality and reflexivity (and also transitivity). Examples were not covered as a rule as we tried to construct a more general mechanism of finding appropriate models for approximation spaces in Mizar providing more automatization than it is now [10]. Of course, we can see some more general applications of some registrations of clusters, essentially not dealing with the notion of an approximation: the notions of an alliance binary relation were not defined in the Mizar Mathematical Library before, and we should think about other properties which are also absent but needed in the context of rough approximations [9], [5]. Via theory merging, using mechanisms described in [6] and [7], such elementary constructions can be extended to other frameworks.

scholarly journals £-Single Valued Extremally Disconnected Ideal Neutrosophic Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 53
Author(s):  
Fahad Alsharari
Keyword(s):  
Topological Spaces ◽  
Neutrosophic Sets ◽  
Extremally Disconnected ◽  
Fundamental Properties ◽  
Separation Axioms ◽  
New Concepts ◽  
Definition Of

This paper aims to mark out new concepts of r-single valued neutrosophic sets, called r-single valued neutrosophic £-closed and £-open sets. The definition of £-single valued neutrosophic irresolute mapping is provided and its characteristic properties are discussed. Moreover, the concepts of £-single valued neutrosophic extremally disconnected and £-single valued neutrosophic normal spaces are established. As a result, a useful implication diagram between the r-single valued neutrosophic ideal open sets is obtained. Finally, some kinds of separation axioms, namely r-single valued neutrosophic ideal-Ri (r-SVNIRi, for short), where i={0,1,2,3}, and r-single valued neutrosophic ideal-Tj (r-SVNITj, for short), where j={1,2,212,3,4}, are introduced. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

scholarly journals Two Types of Intuitionistic Fuzzy Covering Rough Sets and an Application to Multiple Criteria Group Decision Making

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 462 ◽  
Author(s):  
Jingqian Wang ◽  
Xiaohong Zhang
Keyword(s):  
Decision Making ◽  
Rough Set ◽  
Group Decision Making ◽  
Rough Sets ◽  
Group Decision ◽  
Multiple Criteria ◽  
Approximation Spaces ◽  
Intuitionistic Fuzzy ◽  
Definition Of ◽  
Covering Rough Set

Intuitionistic fuzzy rough sets are constructed by combining intuitionistic fuzzy sets with rough sets. Recently, Huang et al. proposed the definition of an intuitionistic fuzzy (IF) β -covering and an IF covering rough set model. In this paper, some properties of IF β -covering approximation spaces and the IF covering rough set model are investigated further. Moreover, we present a novel methodology to the problem of multiple criteria group decision making. Firstly, some new notions and properties of IF β -covering approximation spaces are proposed. Secondly, we study the characterizations of Huang et al.’s IF covering rough set model and present a new IF covering rough set model for crisp sets in an IF environment. The relationships between these two IF covering rough set models and some other rough set models are investigated. Finally, based on the IF covering rough set model, Huang et al. also defined an optimistic multi-granulation IF rough set model. We present a novel method to multiple criteria group decision making problems under the optimistic multi-granulation IF rough set model.

scholarly journals Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1274
Author(s):  
Irina Perfilieva ◽  
Ahmed A. Ramadan ◽  
Enas H. Elkordy
Keyword(s):  
Computer Science ◽  
Fuzzy Systems ◽  
Topological Spaces ◽  
Fuzzy Relations ◽  
Approximation Spaces ◽  
Galois Correspondence ◽  
Closure Spaces

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

On the union and intersection operations of rough sets based on various approximation spaces

2015 ◽  
Vol 292 ◽  
pp. 214-229 ◽  
Author(s):  
Xiaohong Zhang ◽  
Jianhua Dai ◽  
Yucai Yu
Keyword(s):  
Rough Sets ◽  
Approximation Spaces

Regular $${G_\delta}$$ G δ -diagonals and some upper bounds for cardinality of topological spaces

2016 ◽  
Vol 149 (2) ◽  
pp. 324-337
Author(s):  
I. S. Gotchev ◽  
M. G. Tkachenko ◽  
V. V. Tkachuk
Keyword(s):  
Upper Bounds ◽  
Topological Spaces ◽  
Delta G
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