On Some New Types of Fuzzy Soft Compact Spaces

Saleh, S.; Al-shami, Tareq M.; Mhemdi, Abdelwaheb

doi:https://doi.org/10.1155/2023/5065592

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Abstract Introduction Results Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5065592 | https://doi.org/10.1155/2023/5065592

On Some New Types of Fuzzy Soft Compact Spaces

S. Saleh,^1,2Tareq M. Al-shami,^3,4and Abdelwaheb Mhemdi⁵

Academic Editor: Ljubisa Kocinac

Received20 Sept 2022

Revised02 Dec 2022

Accepted06 Feb 2023

Published20 Feb 2023

Abstract

In this article, we apply the concept of fuzzy soft -open sets to define new types of compactness in fuzzy soft topological spaces, namely, -compactness and -compactness. We explore some of their basic properties and reveal the relationships between these types and that which are defined by other authors. Also, we show that -compactness is more general than that which is presented in other papers. To clarify the obtained results and relationships, some illustrative examples are given.

1. Introduction

Conventional mathematical tools are not sufficient to cope with many practical problems that existed in various disciplines such as social science, economics, engineering, medical science, and environment, involving vagueness/uncertainty. Zadeh [1] proposed the remarkable theory of fuzzy set (F-set) for handling these kinds of uncertainties where traditional tools fail. This theory represents a real revolution and grand paradigmatic change in mathematics. Since it is advent, it has contributed to handling numerous types of practical issues and overcoming many real-life problems. On the other hand, this theory has its inherent difficulties, which are possibly attributed to the inadequacy of the parameterization tool, as Molodtsov pointed out in [2]. So, he familiarized the idea of the soft set (S-set) as an efficient instrument for coping with uncertainties that are free from the abovementioned difficulties. Shabir and Naz [3] studied the topological structures of S sets. Then, many contributions have been conducted to soft topologies by many authors; see for example [4–6].

In recent times, the fuzzification of the S-set is progressing rapidly combining F sets and S sets, Maji et al. [7] established a novel model known as a fuzzy soft set (FS-set). The important notion of mapping FS classes was studied by Kharal and Ahmad [8]. Subsequently, the topological structure of FS sets was started by Tanay and Kandemir [9]. FS sets and their applications have been studied in different aspects, such as algebraic ([10–12]) and decision-making ([13, 14]). Mukherjee et al. [15] introduced the notions of fuzzy soft -open (closed) sets and fuzzy soft -closure (interior) with fuzzy soft -continuous mappings. The notion of compactness in fuzzy soft topological space (FSTS) was defined and studied in ([16, 17]) as a generalization of Chang’s notion [18]. Taha [19] discussed compactness via the structure of fuzzy soft and r-Minimal spaces. Also, Saleh and Salemi [20] defined and studied a general type of fuzzy soft compactness.

In this work, we define and study new types of compactness in FSTS, namely, -compactness and -compactness. Some of their basic properties are studied. The relationships between these notions and that which are defined by other authors are investigated. We show that -compactness is more general than that which are presented in [16, 17, 20]. Some results and theorems related to these notions are studied with some necessary examples.

2. Basic Definitions and Results

Here, we recall the basic definitions which will be needed in this paper. refers to the universal set, is the set of all parameters for , -refers to fuzzy soft, means an FSTS, and .

Definition 1 (see [1, 18]). An F-set of is a mapping and can be represented by the pairs . refers to the set of all F sets on . An F-point, is an F-set in given by at and otherwise for all . For , refers to the F-constant function where for all .

Definition 2 (see [21]). An FS-set over symbolized by is the set of ordered pairs . refers to the set of all FS sets over . The complement of symbolized by , where defined by . Clearly, . , , is called an FS-constant set.

Definition 3 (see [7, 21]). Let be two FS sets on . Then,(1) is called a null (resp universal) FS-set, symbolized by if for all (2) is a subset of if , symbolized by (3) and are equal if and . It is symbolized by (4)The union of and is an FS-set defined by for all . is symbolized by .(5)The intersection of and is an FS-set defined by for all . is symbolized by

Definition 4 (see [22–24]). An FS-point over is an FS-set over defined by, if and if , where is the F-point in . refers to the set of all FS points in . An FS-point is said to belong to an FS-set, symbolized by if . Moreover, every non-nullFS-set can be expressed as the union of all the FS points belonging to .

Definition 5 (see [9, 22]). A triplet is FSTS, where is a family of FS sets on which satisfies the following conditions:(1) and belong to (2) is closed under arbitrary union and finite intersection

Definition 6 (see [22, 24, 25]). The FS sets and are called quasi-coincident, symbolized by q if there exist , such that, . If is not quasi-coincident with , then we write .

Definition 7 (see [15, 24]). An FS-set of is called Q-neighborhood (briefly, q-nbd) of if there exists a fuzzy soft open set (FSO-set) such that .

Definition 8 (see [8]). Let and be two families of all FS sets over and , respectively. Let and be two maps. Then, is called a fuzzy soft map (FS-map), for which(i)If , then the image of symbolized by is an FS-set on given by , and otherwise, for all (ii)If , then the preimage of symbolized by is an FS-set on defined by, for all An FS-map is called one-to-one(onto), if and are one-to-one(onto).

Theorem 1 (see [8]). Let andfor all , where is an index set. Then, for an FS-map, we have(1)if, then(2)if, then(3)(4), the equality holds if is one-to-one(5)(6)(7) if is onto(8)

Definition 9 (see [15, 26]). For an FS-set in , we have(i)An FS-closure of is the intersection of all FSC sets containing , symbolized by , and an FS-interior of is the union of all FSO sets contained in , symbolized by (ii) is said to be a fuzzy soft regular open set (FSRO-set) if and the complement of an FSRO-set is called a fuzzy soft-regular closed set (FSRC-set). denotes the set of all FSRO sets, and denotes the set of all FSRC sets(iii) is said to be an FS semiopen set if and the complement of an FS semiopen set is called an FS semiclosed set(iv) is said to be an -neighborhood (briefly, -nbd) of if and only if there exists FSRO q-nbd (FSRO q-nbd) of , such that

Definition 10 (see [15]). Let be an FS-set in . Then,(i)An FS-point is called an -cluster point of if and only if every FSRO q-nbd of , . The set of all FS -cluster points of is called the FS -closure of , symbolized by , that is, .(ii)An FS-set is called an FS -closed set (FS -set, briefly) if and only if and the complement of an FS -closed set is called an FS -open set(FS -set, briefly). denotes to the set of all sets, and the set of all sets is symbolized by .(iii)The -interior of is symbolized by and defined by , that is, . Consequently, is -set if and only if .

Result 1. [15] Every FSRO-set is an -set, and every -set is an FSO-set. Moreover, if is an FS semiopen set in , then .

Result 2. [15] If is an FSO-set in , then is an FSRC-set, that is, , and for any FS-set in , .

Theorem 2 (see [15]). For FS sets and in we have(1) and (2)If , then (3) is an -set, that is, (4)If are sets, then is an -set

Result 3. (Remark 4.15, [15]) The FS -closure operator on satisfies the Kuratowski closure axioms. So that, there exists one topology over , this topology is defined as follows:
The set of all sets of forms FST, symbolized by . It is called -topology over , and the triplet is called an -topological space. Moreover, .

Definition 11 (see [15]). Let be an FS-map. Then, is -open (-closed) if is -set(-set) in for every -set(-set) in .

Theorem 3 (see [15]). Let be an FS-map. Then, the following items are equivalence:(1) is -continuous(2) is an -set in for every -set in (3) is an -set in for every -set in

Definition 12 (see [22, 27]). (i)Let be an FTS. Then, the collection defines FST on (ii)Let be an FSTS. Then, the collection , for every defines FT on

3. New Types of Fuzzy Soft Compact Spaces

In this section, we are going to give the definitions of two types of FS-compactness, namely, -compactness and -compactness via sets, and study some properties of them.

Definition 13. Let be a family of sets of and be an FS-set over . Then,(i) is said to be a fuzzy soft -open cover (-open cover, briefly) of if . A finite subcover is a finite subfamily of , which is again an -open cover(ii) is said to be a fuzzy soft -open cover (-open cover, briefly) of if for every there is satisfying

Remark 1. Clearly, every -open cover is an -open cover. The converse is not necessarily true. This follows from the fact that an FSO-set does not imply an FS -open set.

Remark 2. Clearly, every -open cover is an -open cover. The converse is not necessarily true, the next Example 1 shows it.

Definition 14. Let be an FS-set in . Then,(i) is said to be an FS -compact if every -open cover of has a finite subcover. is said to be fuzzy soft -compact (-compact, briefly) if itself is -compact(ii) is said to be an FS -compact if every -open cover of has a finite subcover

Example 1. Let and . Then, the family is FST on , whereIt is clear that, , and for any in , . Since , , and , then . So, . Therefore, the only -open cover of is . So, is an -compact, and also is -compact.

Proposition 1. Every -compact space is -compact.

Proof. It follows directly from Remark 2.
The next example shows that the converse is not true in general.

Example 2. Let be an infinite set, and be two families of FS sets on , given by and for all , . We consider FST on which is generated by the subbase . Then, for all . So, every is FS -open. Similarly, every is FS -open. Since the family is a -open cover of which has no finite subcover, then is not -compact.
On the other hand, is not -open cover of . Indeed, for , there is no FS-open set in such that , so the only -open cover of is the family which has as a finite subcover. Therefore, is -compact.

Proposition 2. If and are finite sets, then every FSTS is -compact (-compact).
The converse is not necessarily true, as the next example shows.

Example 3. Let be an infinite set and . Clearly, is an FST on . One can check that is -compact, while is an infinite set.

Remark 3. For any infinite FS-set in the discrete FSTS, is not -compact (not -compact). Indeed, let be the discrete FS-space. Then, is a family of FSO sets. Clearly, for all . So, every is FS -open. Since , then the family is an -open cover of which has no finite subcover. So, is not -compact. When , then is not -compact.

Proposition 3. If is -compact (-compact) and , then is -compact (-compact).

Proof. Obvious.

Theorem 4. If is -compact and is an -set of , then is -compact.

Proof. Let be an FS -set in and be an arbitrary -open cover of . Since is FS -set, then is -open cover of and is -compact, then has a finite subcover, say , that is, . However, are disjoint. Thus, and so, the result holds.

Corollary 1. Every -closed subspace of -compact is -compact.

Theorem 5. Let be an -compact and be an -set in . Then, is an -compact.

Proof. A similar technique given in Theorem 4 is followed.

Definition 15 (see [20]). Let be a family of FS sets and be an FS-set. Then,(i) is said to behave fuzzy soft Q-intersection (-intersection, briefly) with respect to if there exists with (ii) is called has a fuzzy soft finite intersection property if every finite subfamily of has -intersection The next theorem gives a nice characterization of -compact space.

Theorem 6. is an -compact if and only if every family of sets on , having the has -intersection .

Proof. Let be -compact. Then is -compact and let be the family of sets over , which has the . Suppose that, has no -intersection . Then, for all there is such that . Thus, is an -open cover of . Since is -compact, there is a finite subcover of say, . Hence has no -intersection . This contradicts that has . Then, the result holds.
Conversely, let be an -open cover of . Then, has no -intersection . Thus, has no . So, there are , such that has no -intersection . Then, is a finite subcover of . Then, is -compact. The result holds.

Definition 16 (see [16, 17]). A family of FS sets is called has the FS-finite intersection property if the intersection of the members of each finite subfamily of is not the null FS-set.
In the next result, we characterize an -compact space using FS sets.

Theorem 7. is -compact if and only if every family of sets on , which has the FS-finite intersection property has a non-null intersection.

Proof. Let be -compact and let be the family of FSsets in which has the FS-finite intersection property. To prove that is non-null, suppose otherwise, i.e., , then , that is, is an -open cover of . By compactness, has a finite subcover of say, . This implies that . This contradicts that the family has the FS-finite intersection property. So the result holds.
Conversely, let be an -open cover of . Then, is the family of FS -closed sets in . Since is an -open cover of , then . Thus, has no finite intersection property, that is, there is a finite subset of such that . This implies that is a finite subcover of . So the result holds.
We close this section by discussing the image and preimage of -compactness under -continuous maps.

Theorem 8. If is -compact and is -continuous and onto, then, is -compact.

Proof. Let be an -open cover of in . Since is -continuous. Then, the family is an -open cover of in and is -compact, there is a finite subfamily of say, covers , i.e., . Thus,Since is onto, then . Hence, is -compact.

Theorem 9. Let be -continuous. If is -compact in , then is -compact in .

Proof. Let be an -open cover of . Since is -continuous, is -open in for all . Thus, the family is an -open cover of in . Since is -compact, there is a finite subfamily of say covers , i.e., . Thus,Hence, is -compact.

Theorem 10. Let : be an FS -continuous and one-to-one map. If is -compact in , then is -compact in .

Proof. The proof is similar to that in the above theorem.

4. The Relations among Different Types of Fuzzy Soft Compactness

In this section, we investigate the relations between our notions of FS-compactness in this paper and those in other papers; we make a comparison between our types of FS-compactness and that given in ([16, 17, 20]). First, we mention that Osmanoglu and Tokat [17] and Hussain [16] gave the same definition of FS-compactness as a generalization of Chang’s notion [18]. Their definitions are as follows.

Definition 17 (see [16, 17]). (i)A family of FS sets in is a cover of FS-set if . It is open cover if each member of is an FSO-set. A subcover of is a subfamily of which is also a cover.(ii)An FS-set is called a fuzzy soft compact (briefly, -compact) if each FS-open cover of has a finite subcover. is called an -compact if every FS-open cover of has a finite subcover.Saleh and Salemi [20] defined another type of FS-compactness which is more general than that of the above definition. His definition is as follows.

Definition 18. (i)A family of FSO sets is called a fuzzy soft P-open cover (briefly, -open cover) of an FS-set if for every there is such that . A finite p-subcover of is a finite subfamily of which is also an -open cover.(ii)An FS-set is called a fuzzy soft -compact (briefly, -compact) if every -open cover of has a finite subcover. is called an -compact if every -open cover of has a finite p-subcover.

Remark 4. Clearly, every -cover is an FS-cover, but the converse may not be true, as shown by Example (3.3) in [20].

Theorem 11. For finite, -compact -compact.

Proof. Let be an FS-compact and be an FS -open cover of . Since every -set is FS-open, thus, is an FS-open cover of . Since is FS-compact, there is a finite subset of such that . Hence, is -compact.

Corollary 2. -compact -compact.

Proof. The proof follows by using the next example.
Let and . Then, the family is an FST on , where is defined as follows:Since the only -open cover of is , then is -compact.
On the other hand, is not FS-compact. Indeed, the family is an FS-open cover of which does not have a finite subcover.
From Theorem 11 and Proposition 1, we obtain the next result.

Result 4. For finite, -compact -compact -compact.

Remark 5. The converse in the implications of the above result may not be true. Corollary 2 and Example 2 show it.

Theorem 12. For finite, -compact -compact.

Proof. Let be an -compact and be a family of sets of which is an -open cover of . It is clear that is also, a family of FSO sets, which is an -open cover of . Since is an -compact, then there is a finite subset of such that . The result holds.

Corollary 3. -compact -compact.

Proof. It follows directly from the fact that -open cover need not imply -open cover, indeed FSO sets need not sets.
In the next, we show that the notions of -compact and -compact are independent.

Corollary 4. -compact -compact

Proof. Example 1, shows that is an -compact, but it is not -compact, indeed the family is an -open cover of which has no finite p-subcover.

Corollary 5. -compact -compact

Proof. Example 2, shows that is an -compact, but it is not -compact, indeed the family is an -open cover of which has no finite subcover.

Proposition 4. For finite, -compact -compact.

Proof. It follows directly from Remark 4.

Corollary 6. -compact -compact.

Proof. It follows by using the next example.
Let be an infinite set and . Then, the family is an FST on , where given by, . Since the only -open cover of is , then is an -compact. However, is not -compact, indeed for the family is -open cover of which has no finite subcover.

Remark 6. From the previous results, we can summarize the relationships among different types of FS-compactness as in the next diagram.

In the following, we study the relation between -compact spaces and some induced fuzzy compact spaces.

Theorem 13. For finite and an FTS , we have, is -compact is fuzzy -compact.

Proof. Let be an -compact and . Let is an arbitrary fuzzy -open cover of . Then, is an -open cover of . Since is an -compact and so, has a finite subcover say, . Thus, the family is a finite subcover of . Hence, is a fuzzy -compact.
Conversely, let be a fuzzy -compact and be an -open cover of in , then the family is a fuzzy -open cover of . Since is a fuzzy -compact, then has a finite subcover say, . Thus, has a finite subcover say, . Hence, is an -compact.

Theorem 14. For a finite set with . If is fuzzy -compact , then is -compact.

Proof. Let be fuzzy -compact and . Let be an arbitrary -open cover of in . Then, is a fuzzy -open cover of . Since is fuzzy -compact , then is fuzzy -compact. Thus, has a finite subcover, say . Therefore, the subfamily, of is a finite subcover. The proof is complete.
The converse of the above theorem may not be true in general. This fact is demonstrated by the next example.

Example 4. Let be an infinite set, and let the family . Then, one can check that is an FST on . Since the only -open cover of is , then is an -compact. On the other hand, , where is the discrete FT on which is not fuzzy -compact, indeed the family is a fuzzy open cover of which has no finite subcover.

Theorem 15. is -compact if and only if every family of sets on , which has the FS-finite intersection property has a non-null intersection.

Proof. In a similar way to that in Theorem 7, from the above theorem and Theorem 7, we obtain the next result.

Corollary 7. is -compact is -compact.

5. Conclusion

In this work, we have defined and studied new types of compactness in fuzzy soft topological spaces, namely, -compactness and -compactness. Some of their basic properties have been explored, and the relationships between these notions and that which were defined by other authors have been investigated. We have shown that -compactness is more general than that which are presented in [16, 17, 20]. Some results and theorems related to these notions have been discussed with some necessary examples. In the future work, we will study the relationships between the concepts given herein and separation axioms via fuzzy soft -open sets. Also, we will construct a general frame of fuzzy soft topology inspired by the class of -Fuzzy soft sets [28]. Through this frame, it can be reintroduced all concepts of fuzzy soft topological spaces and explored their properties and characterizations.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study is supported via funding from Prince Sattam Bin Abdulaziz University project number (PSAU/2023/R/1444).

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
D. Molodtsov, “Soft set theory-First results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.
View at: Publisher Site | Google Scholar
M. Shabir and M. Naz, “On soft topological spaces,” Computers & Mathematics with Applications, vol. 61, no. 7, pp. 1786–1799, 2011.
View at: Publisher Site | Google Scholar
S. A. Ghour, “Soft minimal soft sets and soft prehomogeneity in soft topological spaces,” International Journal of Fuzzy Logic and Intelligent Systems, vol. 21, no. 3, pp. 269–279, 2021.
View at: Publisher Site | Google Scholar
T. M. Al-shami, “Soft somewhat open sets: soft separation axioms and medical application to nutrition,” Computational and Applied Mathematics, vol. 41, no. 5, 2022.
View at: Publisher Site | Google Scholar
T. M. Al-shami, “Compactness on soft topological ordered spaces and its application on the information system,” Journal of Mathematics, vol. 2021, Article ID 6699092, 12 pages, 2021.
View at: Publisher Site | Google Scholar
P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001.
View at: Google Scholar
A. Kharal and B. Ahmad, “Mappings on fuzzy soft classes,” Advances in Fuzzy Systems, vol. 2009, Article ID 407890, 6 pages, 2009.
View at: Publisher Site | Google Scholar
B. Tanay and M. B. Kandemir, “Topological structures of fuzzy soft sets,” Computers & Mathematics with Applications, vol. 61, pp. 412–418, 2011.
View at: Google Scholar
H. Aktas and N. Cagman, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007.
View at: Publisher Site | Google Scholar
B. Ahmad and A. Kharal, “On fuzzy soft sets,” Advances in Fuzzy Systems, vol. 2009, Article ID 586507, 6 pages, 2009.
View at: Publisher Site | Google Scholar
A. Aygünoǧlu and H. Aygun, “Introduction to fuzzy soft groups,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1279–1286, 2009.
View at: Publisher Site | Google Scholar
F. Feng, Y. B. Jun, X. Liu, and L. F. Li, “An adjustable approach to fuzzy soft set based decision making,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 10–20, 2010.
View at: Publisher Site | Google Scholar
A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007.
View at: Publisher Site | Google Scholar
P. Mukherjee, R. P. Chakraborty, and C. Park, “On fuzzy soft -open sets and fuzzy soft -continuity,” Annals of Fuzzy Mathematics and Informatics, vol. 11, no. 2, pp. 327–340, 2015.
View at: Google Scholar
S. Hussain, “On fuzzy soft compact and fuzzy soft locally compact spaces,” Italian Journal of Pure and Applied Mathematics, vol. 42, pp. 526–538, 2019.
View at: Google Scholar
I. Osmanoglu and O. Tokat, “Compact fuzzy soft spaces,” Annals of Fuzzy Mathematics and Informatics, vol. 7, no. 1, pp. 45–51, 2014.
View at: Google Scholar
C. L. Chang, “Fuzzy topological spaces,” Journal of Mathematical Analysis and Applications, vol. 24, no. 1, pp. 182–190, 1968.
View at: Publisher Site | Google Scholar
I. M. Taha, “Compactness on fuzzy soft r-Minimal spaces,” International Journal of Fuzzy Logic and Intelligent Systems, vol. 21, no. 3, pp. 251–258, 2021.
View at: Publisher Site | Google Scholar
S. Saleh and A. Salemi, “Amani: on compactness in fuzzy soft topological spaces,” Sohag Journal of Mathematics, vol. 5, no. 2, pp. 1–5, 2018.
View at: Google Scholar
S. Roy and T. K. Samanta, “A note on fuzzy soft topological spaces,” Annals of Fuzzy Mathematics and Informatics, vol. 3, no. 2, pp. 305–311, 2011.
View at: Google Scholar
S. Atmaca and I. Zorlutuna, “On fuzzy soft topological spaces,” Annals of Fuzzy Mathematics and Informatics, vol. 5, no. 2, pp. 377–386, 2013.
View at: Google Scholar
İ. Demir and O. B. Özbakır, “Some properties of fuzzy soft proximity spaces,” The Scientific World Journal, vol. 2015, Article ID 752634, 10 pages, 2015.
View at: Publisher Site | Google Scholar
B. P. Varol, A. Aygünoğlu, and H. Aygün, “Neighborhood structures of fuzzy soft topological spaces,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 4, pp. 2127–2135, 2014.
View at: Publisher Site | Google Scholar
S. Saleh and A. Salemi, “Amani: on separation axioms in fuzzy soft topological spaces,” South Asian journal of Math, vol. 8, no. 2, pp. 92–102, 2018.
View at: Google Scholar
T. Neog, D. K. Sut, and G. C. Hazarika, “Fuzzy soft topological spaces,” International Journal of Latest Trends in Mathematics, vol. 2, pp. 54–67, 2012.
View at: Google Scholar
S. Saleh, M. H. Shukur, and A. Al-Salemi, “The relationships for some topological structures via fuzzy, soft, and fuzzy soft sets,” in Proceedings of the 4th International Conference on Communication Engineering and Computer Science (CIC-COCOS 2022), pp. 30-31, Erbil,Kurdistan-Iraq, March 2022.
View at: Google Scholar
T. M. Al-shami, J. C. R. Alcantud, and A. Mhemdi, “New generalization of fuzzy soft sets: -Fuzzy soft sets,” AIMS Mathematics, vol. 8, no. 2, pp. 2995–3025, 2022.
View at: Google Scholar

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Copyright © 2023 S. Saleh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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