Fuzzy roughness via ideals

2020 ◽  
Vol 39 (5) ◽  
pp. 6869-6880
Author(s):  
S. H. Alsulami ◽  
Ismail Ibedou ◽  
S. E. Abbas
Keyword(s):  
Fuzzy Set ◽  
Closure Operator ◽  
Approximation Space ◽  
Approximation Spaces ◽  
Fuzzy Connectedness ◽  
Fuzzy Approximation ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Interior Operator

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

2020 ◽  
Vol 16 (02) ◽  
pp. 291-304
Author(s):  
Sutapa Mahato ◽  
S. P. Tiwari
Keyword(s):  
Lattice Structure ◽  
Approximation Space ◽  
Double Negation ◽  
Approximation Spaces ◽  
Bijective Correspondence ◽  
Fuzzy Approximation ◽  
Fuzzy Transformation ◽  
Double Negation Law ◽  
Transformation Systems ◽  
The Relationship

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.

scholarly journals On induced map f# in fuzzy set theory and its applications to fuzzy continuity

2021 ◽  
Author(s):  
Sandeep Kaur ◽  
Nitakshi Goyal
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Closure Operator ◽  
Topological Spaces ◽  
Continuous Mappings ◽  
New Vision ◽  
Saturated Sets ◽  
Interior Operator ◽  
Fuzzy Topological Spaces

Abstract In this paper, we introduce # image of a fuzzy set which gives a induced map f # corresponding to any function f : X → Y , where X and Y are crisp sets. With this, we present a new vision of studying fuzzy continuous mappings in fuzzy topological spaces where fuzzy continuity explains the term of closeness in the mathematical models. We also define the concept of fuzzy saturated sets which helps us to prove some new characterizations of fuzzy continuous mappings in terms of interior operator rather than closure operator.

scholarly journals On Single-Valued Neutrosophic Closure Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1508
Author(s):  
Fahad Alsharari
Keyword(s):  
Closure Operator ◽  
Neutrosophic Set ◽  
Approximation Space ◽  
Closure Operators ◽  
Continuous Mappings ◽  
Closure Spaces ◽  
Boundary Sets ◽  
Interior Operator

This paper aims to mark out new terms of single-valued neutrosophic notions in a Šostak sense called single-valued neutrosophic semi-closure spaces. To achieve this, notions such as β£-closure operators and β£-interior operators are first defined. More precisely, these proposed contributions involve different terms of single-valued neutrosophic continuous mappings called single-valued neutrosophic (almost β£, faintly β£, weakly β£) and β£-continuous. Finally, for the purpose of symmetry, we define the single-valued neutrosophic upper, single-valued neutrosophic lower and single-valued neutrosophic boundary sets of a rough single-valued neutrosophic set αn in a single-valued neutrosophic approximation space (F˜,δ). Based on αn and δ, we also introduce the single-valued neutrosophic approximation interior operator intαnδ and the single-valued neutrosophic approximation closure operator Clαnδ.

scholarly journals Homomorphisms between Fuzzy Approximation Spaces Based on Residuated Lattice

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yuan Zhao
Keyword(s):  
Equivalence Relation ◽  
Residuated Lattice ◽  
Approximation Space ◽  
Approximation Spaces ◽  
Structure Preserving ◽  
Fuzzy Approximation ◽  
Complete Residuated Lattice ◽  
Factor Set

Two kinds of homomorphisms of fuzzy approximation spaces based on complete residuated lattice are proposed. The homomorphisms are structure-preserving maps in some sense. We also introduce the fuzzy approximation subspaces and investigate their correspondence with the homomorphisms. Given a fuzzy equivalence relation, the factor set of the fuzzy approximation space is discussed.

scholarly journals On Characterization of Rough Type-2 Fuzzy Sets

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Tao Zhao ◽  
Zhenbo Wei
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Rough Sets ◽  
Approximation Space ◽  
Fuzzy Approximation ◽  
Approximation Operators ◽  
Sets Theory

Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the generalized Pawlak approximation space are also introduced. Some basic properties of the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators are discussed. The connections between special crisp binary relations and generalized rough type-2 fuzzy approximation operators are further examined. The axiomatic characterization of generalized rough type-2 fuzzy approximation operators is also presented. Finally, the attribute reduction of type-2 fuzzy information systems is investigated.

Measures of uncertainty for a fuzzy probabilistic approximation space

2021 ◽  
pp. 1-24
Author(s):  
Lijun Chen ◽  
Damei Luo ◽  
Pei Wang ◽  
Zhaowen Li ◽  
Ningxin Xie
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fuzzy Relation ◽  
Point Of View ◽  
Approximation Space ◽  
Fuzzy Approximation ◽  
Entropy Measurement ◽  
Fuzzy Relation Matrix ◽  
Relation Matrix

 An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

Order based hierarchies on hesitant fuzzy approximation space

2018 ◽  
Vol 10 (6) ◽  
pp. 1407-1422 ◽  
Author(s):  
Eric C. C. Tsang ◽  
Jingjing Song ◽  
Degang Chen ◽  
Xibei Yang
Keyword(s):  
Approximation Space ◽  
Fuzzy Approximation

scholarly journals Bipolar Fuzzy Implicative Ideals of BCK-Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
G. Muhiuddin ◽  
D. Al-Kadi
Keyword(s):  
Fuzzy Set ◽  
Extension Property ◽  
Fuzzy Ideal

The notion of bipolar fuzzy implicative ideals of a BCK-algebra is introduced, and several properties are investigated. The relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal is studied. Characterizations of a bipolar fuzzy implicative ideal are given. Conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal are provided. Extension property for a bipolar fuzzy implicative ideal is stated.

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