New Approaches of Rough Sets via Ideals

2016 ◽  
pp. 247-264 ◽  
Author(s):  
Ali Kandil ◽  
M. M. Yakout ◽  
A. Zakaria
Keyword(s):  
Rough Sets ◽  
Present Method ◽  
Boundary Region ◽  
Topological Spaces ◽  
Membership Functions ◽  
New Approaches ◽  
Approximation Operators ◽  
Definition Of ◽  
Lower And Upper Approximations

An ideal I on a nonempty set X is a subfamily of P(X) which is closed under finite unions and subsets. In this chapter, a new definition of approximation operators and rough membership functions via ideal has been introduced. The concepts of lower and upper approximations via ideals have been mentioned. These new definitions are comparing with Pawlak's, Yao's and Allam's definitions. It's therefore shown that the current definitions are more generally. Also, it's shown that the present method decreases the boundary region. In addition to these points, the topology generated via present method finer than Allam's one which is a generalization of that obtained by Yao's method. Finally, T1 topological spaces are obtained by relations and ideals which are not discrete.

scholarly journals Topological Structures of Lower and Upper Rough Subsets in a Hyperring

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Nabilah Abughazalah ◽  
Naveed Yaqoob ◽  
Kiran Shahzadi
Keyword(s):  
Rough Sets ◽  
Topological Spaces ◽  
Topological Structures ◽  
Lower And Upper Approximations

In this paper, we study the connection between topological spaces, hyperrings (semi-hypergroups), and rough sets. We concentrate here on the topological parts of the lower and upper approximations of hyperideals in hyperrings and semi-hypergroups. We provide the conditions for the boundary of hyp-ideals of a hyp-ring to become the hyp-ideals of hyp-ring.

scholarly journals Generalization of Pawlak’s Approximations in Hypermodules by Set-Valued Homomorphisms

2017 ◽  
Vol 42 (1) ◽  
pp. 59-81 ◽  
Author(s):  
Saeed Mirvakili ◽  
Seid Mohammad Anvariyeh ◽  
Bijan Davvaz
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Congruence Relation ◽  
Upper Approximation ◽  
Algebraic Sets ◽  
Set Valued Mapping ◽  
Approximation Operators ◽  
Lower And Upper Approximations ◽  
Generalized Rough Set

AbstractThe initiation and majority on rough sets for algebraic hyperstructures such as hypermodules over a hyperring have been concentrated on a congruence relation. The congruence relation, however, seems to restrict the application of the generalized rough set model for algebraic sets. In this paper, in order to solve this problem, we consider the concept of set-valued homomorphism for hypermodules and we give some examples of set-valued homomorphism. In this respect, we show that every homomorphism of the hypermodules is a set-valued homomorphism. The notions of generalized lower and upper approximation operators, constructed by means of a set-valued mapping, which is a generalization of the notion of lower and upper approximations of a hypermodule, are provided. We also propose the notion of generalized lower and upper approximations with respect to a subhypermodule of a hypermodule discuss some significant properties of them.

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 Comparative Studies of Covering Rough Set Models

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Zhaohao Wang
Keyword(s):  
Rough Set ◽  
Comparative Studies ◽  
Rough Sets ◽  
Upper Approximation ◽  
Approximation Operators ◽  
Definition Of ◽  
Covering Rough Set

Many different proposals exist for the definition of lower and upper approximation operators in covering-based rough sets and so many different covering rough set models are built correspondingly. It is meaningful to explore the connection of these covering rough set models for their applications in practice. In this paper, we establish relationships between the most commonly used operators in covering rough set models. We investigate the conditions under which two types of upper approximation operations are identical.

scholarly journals Rough sets theory via new topological notions based on ideals and applications

2021 ◽  
Vol 7 (1) ◽  
pp. 869-902
Author(s):  
Mona Hosny ◽  
◽  
Keyword(s):  
Rough Set Theory ◽  
Boundary Region ◽  
Membership Functions ◽  
Rough Sets Theory ◽  
And Topology ◽  
Accuracy Measure ◽  
Essential Properties ◽  
Lower And Upper Approximations ◽  
Sets Theory ◽  
Better Than

<abstract><p>There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely $ \mathcal{I} $-$ {\delta_{\beta}}_{J} $-open sets and $ \mathcal{I} $-$ {\bigwedge_{\beta}}_{J} $-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.</p></abstract>

A novel connection between rough sets, hypergraphs and hypergroups

2017 ◽  
Vol 09 (04) ◽  
pp. 1750044 ◽  
Author(s):  
Tita Khalis Maryati ◽  
Bijan Davvaz
Keyword(s):  
Equivalence Relation ◽  
Rough Sets ◽  
General Framework ◽  
Approximation Operators ◽  
Lower And Upper Approximations

This paper presents a general framework for the study of the relations between hypergraphs and hypergroups based on approximation operators. Indeed, by a given hypergraph [Formula: see text], an equivalence relation [Formula: see text] on the set of vertices and using the notion of lower and upper approximations, we investigate two new hyperoperations on the set of vertices. In particular, by considering certain conditions on the equivalence relation [Formula: see text], we obtain two related [Formula: see text]-groups and we prove some results in this respect.

Analysing diabetes using rough sets

2020 ◽  
pp. 533-538
Author(s):  
S. Arjun Raj ◽  
M. Vigneshwaran
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
International Diabetes Federation ◽  
Published Data ◽  
Lower And Upper Approximations

In this article we use the rough set theory to generate the set of decision concepts in order to solve a medical problem.Based on officially published data by International Diabetes Federation (IDF), rough sets have been used to diagnose Diabetes.The lower and upper approximations of decision concepts and their boundary regions have been formulated here.

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 Canonical brackets of a toy model for the Hodge theory without its canonical conjugate momenta

2015 ◽  
Vol 30 (20) ◽  
pp. 1550115 ◽  
Author(s):  
D. Shukla ◽  
T. Bhanja ◽  
R. P. Malik
Keyword(s):  
Present Method ◽  
Hodge Theory ◽  
Rigid Rotor ◽  
Normal Ordering ◽  
Creation And Annihilation Operators ◽  
Basic Concepts ◽  
Definition Of ◽  
Continuous Symmetries ◽  
Internal Symmetries ◽  
Theory Lead

We consider the toy model of a rigid rotor as an example of the Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism and show that the internal symmetries of this theory lead to the derivation of canonical brackets amongst the creation and annihilation operators of the dynamical variables where the definition of the canonical conjugate momenta is not required. We invoke only the spin-statistics theorem, normal ordering and basic concepts of continuous symmetries (and their generators) to derive the canonical brackets for the model of a one [Formula: see text]-dimensional (1D) rigid rotor without using the definition of the canonical conjugate momenta anywhere. Our present method of derivation of the basic brackets is conjectured to be true for a class of theories that provide a set of tractable physical examples for the Hodge theory.

scholarly journals Some Results on Multigranulation Neutrosophic Rough Sets on a Single Domain

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 417 ◽  
Author(s):  
Hu Zhao ◽  
Hong-Ying Zhang
Keyword(s):  
Algebraic Structure ◽  
Rough Sets ◽  
Lattice Structure ◽  
Single Domain ◽  
One Dimensional ◽  
Basic Properties ◽  
Approximation Operators ◽  
Rough Approximation ◽  
The One ◽  
Comprehensive Study

As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the one-dimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved.

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