RETRACTED

This article has been retracted by Hindawi following an investigation undertaken by the publisher [1]. This investigation has uncovered evidence of one or more of the following indicators of systematic manipulation of the publication process:

(1)Discrepancies in scope(2)Discrepancies in the description of the research reported(3)Discrepancies between the availability of data and the research described(4)Inappropriate citations(5)Incoherent, meaningless and/or irrelevant content included in the article(6)Peer-review manipulation

The presence of these indicators undermines our confidence in the integrity of the article’s content and we cannot, therefore, vouch for its reliability. Please note that this notice is intended solely to alert readers that the content of this article is unreliable. We have not investigated whether authors were aware of or involved in the systematic manipulation of the publication process.

Wiley and Hindawi regrets that the usual quality checks did not identify these issues before publication and have since put additional measures in place to safeguard research integrity.

We wish to credit our own Research Integrity and Research Publishing teams and anonymous and named external researchers and research integrity experts for contributing to this investigation.

The corresponding author, as the representative of all authors, has been given the opportunity to register their agreement or disagreement to this retraction. We have kept a record of any response received.

Journal of Mathematics
Volume 2021 (2021), Article ID 9965074, 7 pages
https://doi.org/10.1155/2021/9965074

Research Article

Generalization of Fuzzy Soft -Algebras

N. Alam,¹ G. Muhiuddin,² S. Obeidat,¹ H. N. Zaidi,¹ A. Altaleb,¹ and J. M. Aqib³

¹Department of Basic Sciences, Deanship of Preparatory Year, University of Háil, Háil 2440, Saudi Arabia
²Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
³Department of Software Engineering, College of Computer Science, University of Háil, Háil 2440, Saudi Arabia

Correspondence should be addressed to G. Muhiuddin; chishtygm@gmail.com

Received 26 March 2021; Accepted 5 June 2021; Published 24 June 2021

Academic Editor: naeem jan

Copyright © 2021 N. Alam 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.

Abstract.

In this paper, the notions of -fuzzy soft -algebras and -fuzzy soft sub--algebras are introduced, and related properties are investigated. Furthermore, relations between fuzzy soft -algebras and -fuzzy soft -algebras are displayed. Moreover, conditions for an -fuzzy soft -algebra to be a fuzzy soft -algebra are provided. Also, the union, the extended intersection, and the “AND”-operation of two -fuzzy soft (sub-)-algebras are discussed, and a characterization of an -fuzzy soft -algebra is established.

1. Introduction

The uncertainty which appeared in economics, engineering, environmental science, medical science, social science, and so on is too complicated to be captured within a traditional mathematical framework. In order to overcome this situation, a number of approaches including fuzzy set theory [1, 2], probability theory, rough set theory [3, 4], vague set theory [5], and the interval mathematics [6] have been developed. The concept of soft set was introduced by Molodtsov [7] as a new mathematical method to deal with uncertainties free from the errors being occurred in the existing theories. Later, Maji et al. [8, 9] defined fuzzy soft sets and also described how soft set theory is applied to the problem of decision making. Study on the soft set theory is currently moving forward quickly. In [10], Jun et al. discussed the intersection-soft filters in -algberas. Roh and Jun [11] studied positive implicative ideals of -algebras based on intersectional soft sets. Roy and Mayi [12] gave results on applying fuzzy soft sets to the problem of decision making. Aygünoǧlu and Aygün [13] proposed and investigated the notion of a fuzzy soft group. Furthermore, Jun et al. [14] applied the theory of fuzzy soft sets to -algebras and introduced the notion of fuzzy soft -algebras (briefly, FSB-algebras) and related notions. Moreover, Muhiuddin et al. studied and applied the soft set theory to the different algebraic structures on various aspects (see, e.g., [15–23]). Also, some related concepts based on the present work are studied in [24–33].

In this paper, we define the notions of -FSB-algebras and -fuzzy soft sub--algebras. Further, we investigate related properties and consider relations between fuzzy soft -algebras and -fuzzy soft -algebras. Moreover, we prove that every FSB-algebra over is an -FSB-algebra over and also show by an example that the converse of the aforesaid statement is not true in general. In fact, we provide a condition for an -FSB-algebra to be a FSB-algebra. In addition, we discuss the union, the extended intersection, and the “AND”-operation of two -FSB-algebras. Finally, we establish a characterization of an -fuzzy soft -algebra. The paper is organized as follows. Section 2 summarizes some definitions and properties related to -algebras, fuzzy sets, soft sets, and fuzzy soft sets which are needed to develop our main results. In Section 3, the notions of FSB-algebras are studied and the concepts of -identity and -absolute FSB-algebras are introduced. Section 4 is devoted to the study of -FSB-algebra. The paper ends with a conclusion and a list of references.

2. Preliminaries

A -algebra is the most important class of logical algebras which was introduced by K. Iséki.

By a -algebra, we mean a system , where be a nonempty set with a constant 0 and a binary operation if

(i)(ii)(iii)(iv) If a -algebra satisfies(v), then is called a -algebra. Any -algebra satisfies (a1) , (a2) , (a3) , (a4)  where if and only if . The following conditions are satisfied in any -algebra : (a5) . (a6) .

In a -algebra , a nonempty subset of is called a -subalgebra of if .

In a -algebra , a fuzzy set in is called a fuzzy -algebra if it satisfies

In a set , a fuzzy set in of the form

is called a fuzzy point with support and value and is denoted by .

For a fuzzy set in a set and a fuzzy point , Pu and Liu [34] presented the symbol , where . If (resp. ), then we mean (resp. ), and in this case, is said to belong to (resp. be quasi-coincident with) a fuzzy set . If (resp. ), then we mean or (resp. and ).

For an initial universe set and a set of parameters , let denote the power set of and . Molodtsov [7] defined the soft set as follows.

Definition 1. (see [7]). A pair is called a soft set over , where is a function given by

The set for may be considered as the set of -approximate elements of the soft set . Clearly, a soft set is not a set. We refer the reader to [7] for illustration where several examples are presented.

Let denote the set of all fuzzy sets in .

Definition 2. (see [9]). A pair is called a fuzzy soft set over where is a mapping given by

For all , and it is called fuzzy value set of parameter . If , for all , is a crisp subset of , then is degenerated to be the standard soft set. Thus, fuzzy soft sets are a generalization of standard soft sets.

We will use to denote the set of all fuzzy soft sets over .

Definition 3. (see [9]). Let . The union of and is defined to be the fuzzy soft set satisfying the following conditions:

(i),(ii)for all ,

In this case, we write .

Definition 4. (see [9]). If , then “ AND ” denoted by is defined by

where .

Definition 5. (see [35]). For two soft sets and , the extended intersection is the soft set where , and for every ,

We write .

Definition 6. (see [35]). Let such that . The restricted intersection of and is denoted by and is defined as , where and for all , .

3. -Fuzzy Soft -Algebras

Definition 7. (see [36]). A fuzzy set in is said to be an -fuzzy subalgebra of if

Definition 8. Let where . If there exists a parameter such that is an -fuzzy subalgebra of , we say that is an -fuzzy soft -algebra over based on a parameter . If is an -fuzzy soft -algebra over based on all parameters, we say that is an -fuzzy soft -algebra over .

The notion will be used for the set of all -fuzzy soft -algebras.

Example 1. Let be a -algebra with the following table.

Let and let . Then, , , and are fuzzy sets in . We define them as follows:

Then, is an -fuzzy soft -algebra over .

Proposition 1. If , then

where is any parameter in .

Proof. For and , we have

Hence, for all and any parameter in .

Theorem 1. Let . If , then .

Proof. (straightforward).

The following example shows that there exists such that

(i) is not an -fuzzy soft -algebra over (ii)There exists a subset of such that is an -fuzzy soft -algebra over

Example 2. Consider a -algebra with the following table.

Let and let . Then, , , , , and are fuzzy sets in . We define them as follows:

Then, is not an -fuzzy soft -algebra over since it is not an -fuzzy soft -algebra over based on two parameters and . However, if we take , then is described as follows:

and it is an -fuzzy soft -algebra over .

Theorem 2. Every fuzzy soft -algebra over is an -fuzzy soft -algebra over .

Proof. (straightforward).

The converse of Theorem 2 is not true as follows.

Example 3. Consider in Example 1. We know that is not a fuzzy soft -algebra over since is not a fuzzy soft -algebra over based on the parameter as .

Lemma 1. (see [36]). A fuzzy set in is an -fuzzy subalgebra of

According to the Lemma 1, the following theorem is straightforward.

Theorem 3. A fuzzy soft set if and only if

Theorem 4. If such that

then is a fuzzy soft -algebra over .

Proof. Let and . Since , it follows from Theorem 3 and (13) that

Therefore, is a fuzzy soft -algebra over .

Theorem 5. If , then the extended intersection of and is an -fuzzy soft -algebra over .

Proof. Let be the extended intersection of and . Then, . For any , if (resp. ), then (resp. ). If , then for all since the intersection of two -fuzzy -algebras is an -fuzzy -algebra. Therefore, .

Corollary 1. The restricted intersection of two -fuzzy soft -algebras is an -fuzzy soft -algebra.

Theorem 6. Let . If , then the union .

Proof. By Definition 3, we can write , where and for all ,

Since or for all . If , then because . If , then because . Hence, .

The following illustration shows that Theorem 6 is not valid if .

Example 4. Let be a -algebra with the following table.

Consider sets of parameters as follows:

Then, and are not disjoint sets of parameters. Let be a fuzzy soft set over . Then, , , , and are fuzzy sets in . We define them as follows:

Then, . Let . Then, , , and are fuzzy sets in . We define them as follows:

Then, , and the union

of and is described as follows:

For a parameter , we have

Thus, from Theorem 3, based on the parameter and so that .

Theorem 7. If , then .

Proof. By Definition 4, we have

where . For any , we have

Hence, based on by using Theorem 3. Since is arbitrary,

Definition 9. Let . We say that is an -fuzzy soft sub--algebra of if

(1),(2) is an -fuzzy sub--algebra of for all ; that is, is an -fuzzy -algebra satisfying the following condition:

Example 5. Let in Example 1. For a subset of , let be fuzzy soft set over which is defined as follows:

Then, is an -fuzzy soft sub--algebra of .

Example 6. Let be a -algebra with the following Cayley table.

Let be a set of parameters and let which is defined as follows:

Then, . For a subset of , let defined by

Then, is an -fuzzy soft sub--algebra of .

Theorem 8. Let . If for all , then is an -fuzzy soft sub--algebra of .

Proof. (straightforward).

Theorem 9. Let . If and are -fuzzy soft sub--algebras of , then so is the extended intersection of and .

Proof. The proof is followed from Theorem 5 and Definition 9.

Theorem 10. Let . If and are -fuzzy soft sub--algebras of , then so is the union of and whenever and are disjoint.

Proof. The proof is followed from Theorem 6 and Definition 9.

Theorem 11. Let . If and are -fuzzy soft sub--algebras of , then is an -fuzzy soft sub--algebra of .

Proof. The proof is followed from Theorem 7 and Definition 9.

4. Conclusion

In this paper, we introduced the notions of -fuzzy soft -algebras and -fuzzy soft sub--algebras and investigated their related properties. Also, we discussed relations between fuzzy soft -algebras and -fuzzy soft -algebras. Moreover, conditions for an -fuzzy soft -algebra to be a fuzzy soft -algebra are provided. Moreover, the union, the extended intersection, and the “AND”-operation of two -fuzzy soft (sub-) -algebras are discussed, and a characterization of an -fuzzy soft -algebra is established.

We hope that this work will provide a deep impact on the upcoming research in this field and other soft algebraic studies to open up new horizons of interest and innovations. To extend these results, one can further study these notions on different algebras such as rings, hemirings, -semigroups, semihypergroups, semihyperrings, BL-algebras, MTL-algebras, R₀-algebras, MV-algebras, EQ-algebras, d-algebras, Q-algebras, and lattice implication algebras. Some important issues for future work are (1) to develop strategies for obtaining more valuable results and (2) to apply these notions and results for studying related notions in other algebraic (soft) structures.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by Research Deanship at the University of Háil, Saudi Arabia, through project number RG-20 189.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.

L. A. Zadeh, “Toward a generalized theory of uncertainty (GTU)--an outline,” Information Sciences, vol. 172, no. 1-2, pp. 1–40, 2005.

Z. A. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341–356, 1982.

Z. Pawlak and A. Skowron, “Rudiments of rough sets,” Information Sciences, vol. 177, no. 1, pp. 3–27, 2007.

W.-L. Gau and D. J. Buehrer, “Vague sets,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 23, no. 2, pp. 610–614, 1993.

M. B. Gorzalzany, “A method of inference in approximate reasoning based on interval-valued fuzzy sets,” Fuzzy Sets and Systems, vol. 21, pp. 1–17, 1987.

D. Molodtsov, “Soft set theory-First results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.

P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002.

P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” International Journal of Fuzzy Mathematics and Systems, vol. 9, no. 3, pp. 589–602, 2001.

Y. B. Jun, S. S. Ahn, and K. J. Lee, “Intersection-soft filters in R₀-algebras,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 950897, 7 pages, 2013.

E. H. Roh and Y. B. Jun, “Positive implicative ideals of BCK-algebras based on intersectional soft sets,” Journal of Applied Mathematics, vol. 2013, Article ID 853907, 9 pages, 2013.

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.

A. Aygünoǧlu and H. Aygün, “Introduction to fuzzy soft groups,” Computers & Mathematics with Applications, vol. 58, pp. 1279–1286, 2009.

Y. B. Jun, K. J. Lee, and C. H. Park, “Fuzzy soft set theory applied to BCK/BCI-algebras,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3180–3192, 2010.

A. Al-roqi, G. Muhiuddin, and S. Aldhafeeri, “Normal unisoft filters in R₀-algebras,” Cogent Mathematics, vol. 1, no. 4, pp. 1–9, 2017.

Y. B. Jun, S. Z. Song, and G. Muhiuddin, “Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups,” The Scientific World Journal, vol. 2014, Article ID 467968, 11 pages, 2014.

G. Muhiuddin, “Intersectional soft sets theory applied to generalized hypervector spaces,” Analele Universitatii “Ovidius” Constanta—Seria Matematica, vol. 28, no. 3, pp. 171–191, 2020.

G. Muhiuddin, A. Mahboob, and A. Mahboob, “Int-soft ideals over the soft sets in ordered semigroups,” AIMS Mathematics, vol. 5, no. 3, pp. 2412–2423, 2020.

G. Muhiuddin, A. M. Al-roqi, and S. Aldhafeeri, “Filter theory in MTL-algebras based on uni-soft property,” Bulletin of the Iranian Mathematical Society, vol. 43, no. 7, pp. 2293–2306, 2017.

G. Muhiuddin and A. M. Al-roqi, “Unisoft filters in R₀-algebras,” Journal of Computational Analysis and Applications, vol. 19, no. 1, pp. 133–143, 2015.

G. Muhiuddin, F. Feng, and Y. B. Jun, “Subalgebras of BCK/BCI-algebras based on cubic soft sets,” The Scientific World Journal, vol. 2014, Article ID 458638, 2014.

G. Muhiuddin and A. M. Al-roqi, “Cubic soft sets with applications in BCK/BCI-algebras,” Annals of Fuzzy Mathematics and Informatics, vol. 8, no. 2, pp. 291–304, 2014.

G. Muhiuddin and S. Aldhafeeri, “N -soft p-ideals of BCI-algebras,” European Journal of Pure and Applied Mathematics, vol. 12, no. 1, pp. 79–87, 2019.

M. Akram and B. Davvaz, “Generalized fuzzy ideals of K-algebras,” Journal of Multivalued Valued and Soft Computing, vol. 19, pp. 475–491, 2012.

M. Akram, K. H. Dar, and K. P. Shum, “Interval-valued -fuzzy K-algebras,” Applied Soft Computing, vol. 11, no. 1, pp. 1213–1222, 2011.

M. Akram, “Fuzzy soft Lie algebras,” Journal of Multivalued Logic and Soft Computing, vol. 24, no. 5-6, pp. 501–520, 2015.

D. Al-Kadi and G. Muhiuddin, “Bipolar fuzzy BCI-implicative ideals of BCI-algebras,” Journal of Mathematics, vol. 3, no. 1, pp. 88–96, 2020.

G. Muhiuddin and M. Balamurugan, “Hesitant intuitionistic fuzzy soft b-ideals of BCK-algebras,” Annals of Communications in Mathematics, vol. 3, no. 1, pp. 26–34, 2020.

G. Muhiuddin and B. J. Young, “Sup-hesitant fuzzy subalgebras and its translations and extensions,” Annals of Communications in Mathematics, vol. 2, no. 1, pp. 48–56, 2019.

A. Mahboob and N. M. Khan, “Generalized fuzzy gamma-ideals of ordered gamma-semigroups,” Annals of Communications in Mathematics, vol. 2, no. 2, pp. 91–100, 2019.

T. Senapati and K. P. Shum, Cubic Subalgebras of BCH-Algebras, Annals of Communications in Mathematics, Kingsville, Texas, USA, 2018.

A. F. Talee, M. Y. Abbasi, and A. Basar, “On properties of hesitant fuzzy ideals in semigroups,” Annals of Communication in Mathematics, vol. 3, no. 1, pp. 97–106, 2020.

S. Thongarsa, P. Burandate, and A. Iampan, “Some operations of fuzzy sets in UP-algebras with respect to a triangular norm,” Annals of Communication in Mathematics, vol. 2, no. 1, pp. 1–10, 2019.

P. M. Pu and Y. M. Liu, “Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence,” Journal of Mathematical Analysis and Applications, vol. 76, pp. 571–599, 1980.

M. I. Ali, F. Feng, X. Liu, W. K. Min, and M. Shabir, “On some new operations in soft set theory,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1547–1553, 2009.

Y.-B. Jun, “On (α, β)-fuzzy subalgebras of BCK/ BCI-algebras,” Bulletin of the Korean Mathematical Society, vol. 42, no. 4, pp. 703–711, 2005.