International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(4): 369-377
Published online December 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.4.369
© The Korean Institute of Intelligent Systems
Sang Min Yun , Yeon Seok Eom , and Seok Jong Lee
Department of Mathematics, Chungbuk National University, Cheongju, Korea
Correspondence to :
Seok Jong Lee (sjl@cbnu.ac.kr)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In our previous paper, we proposed a new definition of intuitionistic fuzzy rough sets. In this paper, we propose a topology for redefined intuitionistic fuzzy rough sets and investigate the basic properties of their subspaces, transition spaces, and continuous functions. Moreover, we obtain the adjointness between the categories of fuzzy rough sets and intuitionistic fuzzy rough sets. The results obtained from this new definition differ from those of previous studies.
Keywords: Fuzzy rough set, Intuitionistic fuzzy rough sets, Transition topology, Category
Since Zadeh [1] introduced the notion of fuzzy sets, several researchers have attempted to generalize fuzzy sets using various approaches.
One approach has been to use the concept of intuitionistic fuzzy sets introduced by Atanassov [2]. Unlike fuzzy sets, an intuitionistic fuzzy set provides both a membership degree and a non-membership degree. Many important concepts in general topology have been generalized to intuitionistic fuzzy settings.
The other main approach is to utilize the concept of rough sets introduced by Pawlak [3]. The rough set theory proposes a mathematical approach to imperfect knowledge, which is expressed by the boundary region of a set. The rough set concept can be represented by topological approximations, that is, interior and closure. Zhou et al. [4,5] proposed lower and upper approximations of intuitionistic fuzzy sets. Many researchers, including us, have examined the important properties of intuitionistic fuzzy approximation operators [6–8].
Many attempts have been made with the objective of combining fuzziness and roughness. Dubois and Prade [9] and Nanda and Majumda [10] introduced and discussed the concept of fuzzy rough sets. Coker [11] demonstrated that fuzzy rough sets, as proposed by Nanda and Majumdar [10], are intuitionistic L-fuzzy sets developed by Atanassov [2]. Chakrabarty et al. [12] proposed a fuzziness measure in rough sets. By combining the concepts of intuitionistic fuzzy sets and fuzzy rough sets, Samanta and Mondal [13] proposed the idea of intuitionistic fuzzy rough sets. The topology of intuitionistic fuzzy rough sets was accordingly introduced [14,15]. Bashir et al. [16] studied the topological properties of intuitionistic fuzzy rough sets under different conditions like serial, strongly serial, and left continuity.
The concept of intuitionistic fuzzy rough sets has some advantages with regard to decision-making. Zhan and Sun [17] discussed the rough and precision degrees of covering-based intuitionistic fuzzy rough set models. They introduced an intuitionistic fuzzy rough methodology to the multi-attribute decision-making problem, which is more effective than the previous models. Shanthi [18] developed a decision-making method based on the composition of intuitionistic fuzzy rough matrices on a finite universe.
In light of mathematical theory, however, the properties of the “old” intuitionistic fuzzy rough sets are extremely complicated and inadequate in terms of the extension of intuitionistic properties. Hence, remedying this flaw is critical for expanding related theories. To overcome this weakness, we introduced a new definition of fuzzy rough sets and intuitionistic fuzzy rough sets [19].
In this paper, we introduce a topology for redefined intuitionistic fuzzy rough sets and investigate the basic properties of their subspaces, transition spaces, and continuous functions. In addition, we study the categorical relation between the category of fuzzy rough sets and the category of intuitionistic fuzzy rough sets. The results obtained from this new definition are different from those obtained in previous studies. Some of our results are similar to those of Hazra et al. [15]. However, our results are consistent with the theory of intuitionistic fuzzy sets. The results help us base decision-making on a more solid foundation.
In [10], the definition of fuzzy rough sets was introduced. The paper stated “We shall consider (
where
Furthermore, the complement
where
Unfortunately, if we follow this definition, the double complement of fuzzy rough set
Let
where
The
For any fuzzy rough set
For any fuzzy rough set
So,
If
An IF rough set is an intuitive version of a fuzzy rough set. An IF rough set has a membership degree and a non-membership degree comprising fuzzy rough sets. Therefore, this concept is effective in dealing with systems that have a rough membership degree and rough non-membership degree.
We denote by IFRS(
The IF rough sets
Let
where
Subsequently,
Let
where
Then, the inverse image of an IF rough set under
For the other properties of IF rough sets, refer to [19].
One of the benefits of our new definition is the ability to construct a category of rough sets and a category of IF rough sets, where the double negation of a fuzzy rough set becomes itself. With the old definition of [15], constructing a category is difficult.
In this section, we study the connection between two concepts, that is, IF rough sets and fuzzy rough sets.
Let
Let
Now, we are ready to define functors between two categories. Let
Define
Then
Clearly,
Similarly, we have the following functor.
Define
Then,
Define
Then,
Clearly,
Similarly, we have the following functor.
Define
Then,
As in the following two theorems, we have two adjointnesses between the above functors.
The functor
The following diagram commutes.
Take
The functor
Similar to the above proof.
We then obtained a categorical relation between the category of fuzzy rough sets and the category of IF rough sets.
A basic approach for studying the application of a new mathematical system is topology because the topological structure explains the convergence and neighborhoodness of a system. In this section, we introduce the topology of IF rough sets.
Let
,
for all ,
for all
Then, is called a
The collection of ℱ of all closed IF rough sets of (
Obvious from Definition 4.1.
Let
Let
is the smallest closed IF rough set containing
.
.
.
.
Let (
Let (
The continuous function is characterized by closed sets and closure as follows.
The following statements are equivalent:
is IFR continuous.
, for any IF rough set
(1) ⇒ (2) Let
(2) ⇒ (3) Let
(3) ⇒ (1) Let
Let
Let (
This topology is called the
If , then
Then, ℱ
Let (
Suppose that {
Moreover,
If is continuous and
For any open IF rough set
If is continuous, then is continuous.
Take . Then,
Hence, is continuous.
Hazra et al. [15] defined the subspace topology, but it is, in fact, a new concept different from the subspace topology. Consequently, we suggest a new terminology “transition topology.” Insofar as we take the characteristic function of a crisp set, the transition topology and the subspace topology are the same.
Let
.
If , then .
If for all
Let be a topology of IF rough sets of
If we take
Every member of is called an
Subsequently, ℱ
ℱ
Obvious from Definition 6.1.
Let . Then,
because
By the above remark, we can give the following definition.
The collection
forms a topology of IF rough sets on
If we take
ℱ
In general, and are different. For
If is an indiscrete topology of IF rough sets of
If is a discrete topology, then . Thus, .
Let
Thus, .
Following example shows that, in general
Let
Then,
Let
Then,
Let
Then,
Now,
because
We have . Thus,
Clearly, (
In this paper, we introduced a topology for the redefined intuitionistic fuzzy rough sets and investigated the basic properties of their subspaces and continuous functions. We also introduced the concepts of the first and second transition spaces. We confirmed that these two spaces are different but share the same closed sets. In addition, we obtained the adjointness between the categories of fuzzy rough sets and intuitionistic fuzzy rough sets. All results obtained from this new definition are different from those of previous studies. The new approach of intuitionistic fuzzy rough sets enables us to manipulate them more simply and easily. In the following study, we will focus on expanding the related theories.
No potential conflict of interest relevant to this article was reported.
E-mail: jivesm@naver.com
E-mail: math1518@naver.com
E-mail: sjl@cbnu.ac.kr
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(4): 369-377
Published online December 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.4.369
Copyright © The Korean Institute of Intelligent Systems.
Sang Min Yun , Yeon Seok Eom , and Seok Jong Lee
Department of Mathematics, Chungbuk National University, Cheongju, Korea
Correspondence to:Seok Jong Lee (sjl@cbnu.ac.kr)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In our previous paper, we proposed a new definition of intuitionistic fuzzy rough sets. In this paper, we propose a topology for redefined intuitionistic fuzzy rough sets and investigate the basic properties of their subspaces, transition spaces, and continuous functions. Moreover, we obtain the adjointness between the categories of fuzzy rough sets and intuitionistic fuzzy rough sets. The results obtained from this new definition differ from those of previous studies.
Keywords: Fuzzy rough set, Intuitionistic fuzzy rough sets, Transition topology, Category
Since Zadeh [1] introduced the notion of fuzzy sets, several researchers have attempted to generalize fuzzy sets using various approaches.
One approach has been to use the concept of intuitionistic fuzzy sets introduced by Atanassov [2]. Unlike fuzzy sets, an intuitionistic fuzzy set provides both a membership degree and a non-membership degree. Many important concepts in general topology have been generalized to intuitionistic fuzzy settings.
The other main approach is to utilize the concept of rough sets introduced by Pawlak [3]. The rough set theory proposes a mathematical approach to imperfect knowledge, which is expressed by the boundary region of a set. The rough set concept can be represented by topological approximations, that is, interior and closure. Zhou et al. [4,5] proposed lower and upper approximations of intuitionistic fuzzy sets. Many researchers, including us, have examined the important properties of intuitionistic fuzzy approximation operators [6–8].
Many attempts have been made with the objective of combining fuzziness and roughness. Dubois and Prade [9] and Nanda and Majumda [10] introduced and discussed the concept of fuzzy rough sets. Coker [11] demonstrated that fuzzy rough sets, as proposed by Nanda and Majumdar [10], are intuitionistic L-fuzzy sets developed by Atanassov [2]. Chakrabarty et al. [12] proposed a fuzziness measure in rough sets. By combining the concepts of intuitionistic fuzzy sets and fuzzy rough sets, Samanta and Mondal [13] proposed the idea of intuitionistic fuzzy rough sets. The topology of intuitionistic fuzzy rough sets was accordingly introduced [14,15]. Bashir et al. [16] studied the topological properties of intuitionistic fuzzy rough sets under different conditions like serial, strongly serial, and left continuity.
The concept of intuitionistic fuzzy rough sets has some advantages with regard to decision-making. Zhan and Sun [17] discussed the rough and precision degrees of covering-based intuitionistic fuzzy rough set models. They introduced an intuitionistic fuzzy rough methodology to the multi-attribute decision-making problem, which is more effective than the previous models. Shanthi [18] developed a decision-making method based on the composition of intuitionistic fuzzy rough matrices on a finite universe.
In light of mathematical theory, however, the properties of the “old” intuitionistic fuzzy rough sets are extremely complicated and inadequate in terms of the extension of intuitionistic properties. Hence, remedying this flaw is critical for expanding related theories. To overcome this weakness, we introduced a new definition of fuzzy rough sets and intuitionistic fuzzy rough sets [19].
In this paper, we introduce a topology for redefined intuitionistic fuzzy rough sets and investigate the basic properties of their subspaces, transition spaces, and continuous functions. In addition, we study the categorical relation between the category of fuzzy rough sets and the category of intuitionistic fuzzy rough sets. The results obtained from this new definition are different from those obtained in previous studies. Some of our results are similar to those of Hazra et al. [15]. However, our results are consistent with the theory of intuitionistic fuzzy sets. The results help us base decision-making on a more solid foundation.
In [10], the definition of fuzzy rough sets was introduced. The paper stated “We shall consider (
where
Furthermore, the complement
where
Unfortunately, if we follow this definition, the double complement of fuzzy rough set
Let
where
The
For any fuzzy rough set
For any fuzzy rough set
So,
If
An IF rough set is an intuitive version of a fuzzy rough set. An IF rough set has a membership degree and a non-membership degree comprising fuzzy rough sets. Therefore, this concept is effective in dealing with systems that have a rough membership degree and rough non-membership degree.
We denote by IFRS(
The IF rough sets
Let
where
Subsequently,
Let
where
Then, the inverse image of an IF rough set under
For the other properties of IF rough sets, refer to [19].
One of the benefits of our new definition is the ability to construct a category of rough sets and a category of IF rough sets, where the double negation of a fuzzy rough set becomes itself. With the old definition of [15], constructing a category is difficult.
In this section, we study the connection between two concepts, that is, IF rough sets and fuzzy rough sets.
Let
Let
Now, we are ready to define functors between two categories. Let
Define
Then
Clearly,
Similarly, we have the following functor.
Define
Then,
Define
Then,
Clearly,
Similarly, we have the following functor.
Define
Then,
As in the following two theorems, we have two adjointnesses between the above functors.
The functor
The following diagram commutes.
Take
The functor
Similar to the above proof.
We then obtained a categorical relation between the category of fuzzy rough sets and the category of IF rough sets.
A basic approach for studying the application of a new mathematical system is topology because the topological structure explains the convergence and neighborhoodness of a system. In this section, we introduce the topology of IF rough sets.
Let
,
for all ,
for all
Then, is called a
The collection of ℱ of all closed IF rough sets of (
Obvious from Definition 4.1.
Let
Let
is the smallest closed IF rough set containing
.
.
.
.
Let (
Let (
The continuous function is characterized by closed sets and closure as follows.
The following statements are equivalent:
is IFR continuous.
, for any IF rough set
(1) ⇒ (2) Let
(2) ⇒ (3) Let
(3) ⇒ (1) Let
Let
Let (
This topology is called the
If , then
Then, ℱ
Let (
Suppose that {
Moreover,
If is continuous and
For any open IF rough set
If is continuous, then is continuous.
Take . Then,
Hence, is continuous.
Hazra et al. [15] defined the subspace topology, but it is, in fact, a new concept different from the subspace topology. Consequently, we suggest a new terminology “transition topology.” Insofar as we take the characteristic function of a crisp set, the transition topology and the subspace topology are the same.
Let
.
If , then .
If for all
Let be a topology of IF rough sets of
If we take
Every member of is called an
Subsequently, ℱ
ℱ
Obvious from Definition 6.1.
Let . Then,
because
By the above remark, we can give the following definition.
The collection
forms a topology of IF rough sets on
If we take
ℱ
In general, and are different. For
If is an indiscrete topology of IF rough sets of
If is a discrete topology, then . Thus, .
Let
Thus, .
Following example shows that, in general
Let
Then,
Let
Then,
Let
Then,
Now,
because
We have . Thus,
Clearly, (
In this paper, we introduced a topology for the redefined intuitionistic fuzzy rough sets and investigated the basic properties of their subspaces and continuous functions. We also introduced the concepts of the first and second transition spaces. We confirmed that these two spaces are different but share the same closed sets. In addition, we obtained the adjointness between the categories of fuzzy rough sets and intuitionistic fuzzy rough sets. All results obtained from this new definition are different from those of previous studies. The new approach of intuitionistic fuzzy rough sets enables us to manipulate them more simply and easily. In the following study, we will focus on expanding the related theories.
Sang Min Yun and Seok Jong Lee
International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(2): 129-137 https://doi.org/10.5391/IJFIS.2020.20.2.129