[Retracted] Bipolar Fuzzy Implicative Ideals of BCK-Algebras

Muhiuddin, G.; Al-Kadi, D.

doi:https://doi.org/10.1155/2021/6623907

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article Retraction

!

This article has been Retracted. To view the article details, please click the ‘Retraction’ tab above.

Special Issue

Soft Computing Algorithms Based on Fuzzy Extensions

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 6623907 | https://doi.org/10.1155/2021/6623907

[Retracted] Bipolar Fuzzy Implicative Ideals of BCK-Algebras

G. Muhiuddin¹and D. Al-Kadi²

Academic Editor: Lazim Abdullah

Received11 Dec 2020

Revised27 Dec 2020

Accepted02 Jan 2021

Published27 Jan 2021

Abstract

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.

1. Introduction

Fuzzy sets are characterized by a membership function which associates elements with real numbers in the interval [0, 1] that represents its membership degree to the fuzzy set. Several kinds of fuzzy set extensions have been introduced such as interval-valued, intuitionistic, and bipolar-valued fuzzy sets. The bipolar-valued fuzzy set notion [1] was introduced to treat imprecision as in traditional fuzzy sets, where the degree of membership belongs to the interval [0, 1], and we cannot tell apart unrelated elements from the opposite elements. The extension here enlarges the range of the membership degree from the interval [0, 1] to the interval [−1, 1] to solve such a problem (we refer the reader to [2–4]). The membership degrees which lie in the interval [−1, 1] represent the satisfaction degree to the corresponding property in a fuzzy set and its counter property as follows: having a membership degree in the interval [−1, 0) means that the elements are satisfying implicit counter property, having (0, 1] means that the elements are satisfying the property, and having 0 means that the elements are unrelated to the corresponding property.

The bipolar-valued fuzzification has been used to study different notions in BCK/BCI-algebras such as subalgebras and ideals of BCK/BCI-algebras [5], a-ideals of BCI-algebras [6], and more, see the references [7–10]. Other researches also added their contribution to the study in this field on different branches of algebra in various aspects (see, e.g., [11–26]). Also, some more general concepts on bipolar fuzzy have been studied in [27–31].

Recently, the bipolar fuzzy BCI-implicative ideals of BCI-algebras were studied in [32]. Moreover, new types of bipolar fuzzy ideals of BCK-algebras have been investigated in [33], typically bipolar fuzzy (closed, positive implicative, and implicative) ideals. Moreover, some related concepts on fuzzy sets and their useful generalizations were applied in various algebraic structures (see, e.g., [33–53]).

In this paper, we apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain further results in this manner. Furthermore, we consider the relation of a bipolar fuzzy ideal with a bipolar fuzzy implicative ideal. We provide characterizations of a bipolar fuzzy implicative ideal. Moreover, we display conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal. Finally, we discuss extension property for a bipolar fuzzy implicative ideal.

2. Preliminaries

The basic results on BCK-algebras are given in this section.

By a BCK-algebra, we mean an algebra of type (2, 0) satisfying the axioms:(a1)(a2)(a3)(a4)

We can define a partial ordering by if and only if .

In any BCK-algebra , the following hold:(b1)(b2)(b3)(b4)

Let us consider a subset of a BCK-algebra . We say is an ideal if (c1) , (c2) A nonempty subset of a BCK-algebra is called an implicative ideal of if it satisfies (c1) and (c3)

3. Bipolar Fuzzy Ideals

In the following sections, denotes a BCK-algebra.

For any family of real numbers, we define

Moreover, if , then and are denoted by and , respectively.

For a bipolar fuzzy set , we define negative -cut of and the positive-cut of , respectively, as follows:where . The setis called the -cut of . For every , if , then the setis called the -cut of .

Definition 1. (see [5]). A bipolar fuzzy set in a BCK-algebra is called a bipolar fuzzy ideal of if it satisfies the following assertions:(i)(ii)For any and any bipolar fuzzy set in , we letObviously, . If is a bipolar fuzzy ideal of , then . The following is our question: For a bipolar fuzzy set insatisfying Definition 1 (i), isan ideal of ? The following example provides a negative answer; that is, there exists an element such that is not an ideal of .

Example 1. Let be a set with a Cayley table which is given in Table 1.
Then, is a BCK-algebra. Let be a bipolar fuzzy set in defined by

Then, satisfies Definition 1 (i), and it is not a bipolar fuzzy ideal of becauseand/orThen, is not an ideal of since and , while . Note that is an ideal of .

We give conditions for the set to be an ideal.

Theorem 1. Let . If is a bipolar fuzzy ideal of , then is an ideal of .

Proof. We recall that . Let such that and . Then, , , and . Since is a bipolar fuzzy ideal of , we have from Definition 1 (ii) thatand so, . Therefore, is an ideal of .

Theorem 2. Let be a bipolar fuzzy set in and .(1)If is an ideal of , then satisfies the following implications for all :(2)If satisfies Definition 1(i) and (9), then is an ideal of .

Proof. (1)We assume that is an ideal of for each . We suppose that and for all . Then, and . Since is an ideal of , it follows that , that is, and .(2)We suppose that satisfies Definition 1 (i) and (9). For each , let such that and . Then, , , , and , which imply that and . Using (9), we have and , and so, . Since satisfies Definition 1 (i), it follows that . Therefore, is an ideal of .

Lemma 1. (see [5]). Every bipolar fuzzy ideal of satisfies the following implication:

Proposition 1. For any bipolar fuzzy ideal of , the following are equivalent:(1)(2)

Proof. We assume that condition (2) is valid. Note thatfor all by using (b2), (b3), and (b4). It follows from Lemma 1 thatSo, from (b2) and (2), it follows thatThus, (9) holds. Now, we suppose that (9) is valid. Using (b1), (a3), and (9) with replacing by , we havewhich proves (2).

Proposition 2. (see [5]). A bipolar fuzzy set in is a bipolar fuzzy ideal of if and only if for all , implies and .

As a generalization of Proposition 2, we have the following results.

Theorem 3. If a bipolar fuzzy set in is a bipolar fuzzy ideal of , then for all ,where .

Proof. The proof is by induction on . Let be a bipolar fuzzy ideal of . Lemma 1 and Proposition 2 show that condition (15) is valid for . We assume that satisfies condition (15) for , that is, for all , impliesLet such that . Then,Since is a bipolar fuzzy ideal of , it follows from Definition 1 (ii) thatThis completes the proof.

Now, we consider the converse of Theorem 3.

Theorem 4. Let be a bipolar fuzzy set in satisfying condition (15). Then, is a bipolar fuzzy ideal of .

Proof. Note that for all . It follows from (15) that and for all . Let such that . Then,and so,Hence, by Proposition 2, we conclude that is a bipolar fuzzy ideal of .

4. Bipolar Fuzzy Implicative Ideals

Definition 2. A bipolar fuzzy set in is called a bipolar fuzzy implicative ideal of if both the nonempty negative -cut and the nonempty positive -cut of are implicative ideals of for all .

Example 2. Let be a set in which the operation is defined by Table 2.
Then, is a BCK-algebra. Let satisfy , that is, and . Let be a bipolar fuzzy set in given by

By routine calculations, we know that is a bipolar fuzzy implicative ideal of .

Theorem 5. A bipolar fuzzy setin is a bipolar fuzzy implicative ideal of if and only if it satisfies Definition 1 (i) and the following assertions:

Proof. We suppose that is a bipolar fuzzy implicative ideal of . If or for some , then or , which contradicts the fact. Hence, and . For some , we assume that we have the following relation:Then, and , but . This is not possible; therefore, we havefor all . If for some , then and , but . We reach a contradiction because is an implicative ideal of . Henceforth,for all . Consequently, a bipolar fuzzy implicative ideal satisfies Definition 1 (i) and (21).
Conversely, we suppose that satisfies Definition 1 (i) and (21) and let s.th. and . It is clear that . Let be such that and . Then, and . It follows from (21) thatand so, . Hence, is an implicative ideal of . Similarly, we can show thatfor all , and so, . Therefore, is an implicative ideal of . Consequently, is a bipolar fuzzy implicative ideal of .

Next, we have the following theorems.

Theorem 6. A bipolar fuzzy ideal of is a bipolar fuzzy implicative ideal of if and only if it satisfies Proposition 1 (1).

Proof. Let be a bipolar fuzzy implicative ideal of . If is replaced by in (21), thenwhich is Proposition 1 (1). Conversely, let be a bipolar fuzzy ideal of satisfying Proposition 1 (1). Note thatfor all . Using Lemma 1, we haveIt follows from Definition 1 (ii) and Proposition 1 (1) thatThus, is a bipolar fuzzy implicative ideal of .

Combining Proposition 1 and Theorem 6, we have the following characterization of a bipolar fuzzy implicative ideal.

Theorem 7. Let be a bipolar fuzzy ideal of . Then, it is a bipolar fuzzy implicative ideal of if and only if it satisfies Proposition 1 (2).

Theorem 8. (see [33]). Every bipolar fuzzy implicative ideal is a bipolar fuzzy ideal.

Theorem 9. Let be a bipolar fuzzy set in . Then, is a bipolar fuzzy implicative ideal of if and only if it satisfies Definition 1 (i) and

Proof. We suppose that is a bipolar fuzzy implicative ideal of . Then, is a bipolar fuzzy ideal of by Theorem 8, and so, Definition 1 (i) is true. From Theorem 7, it follows that satisfies Proposition 1 (2). Thus,which proves (31). Conversely, let be a bipolar fuzzy set in satisfying Definition 1 (i) and (31). Then,Thus, is a bipolar fuzzy ideal of . Now, we take in (31) and use (b1) and Definition 1 (i) to getIt follows from Theorem 6 that is a bipolar fuzzy implicative ideal of .

Summarizing the abovementioned results, we have a characterization of a bipolar fuzzy implicative ideal of .

Theorem 10. Let be a bipolar fuzzy set in . Then, the following assertions are equivalent:(1) is a bipolar fuzzy implicative ideal of (2) satisfies Definition 1 (i) and (21)(3) is a bipolar fuzzy ideal of satisfying Proposition 1 (1)(4) is a bipolar fuzzy ideal of satisfying Proposition 1 (2)(5) satisfies Definition 1 (i) and (31)

Theorem 11. Let . If is a bipolar fuzzy implicative ideal of , then is an implicative ideal of .

Proof. We recall that . Let such that and . Then, , , , and . Since is a bipolar fuzzy implicative ideal of , it follows from (21) thatso that . Therefore, is an implicative ideal of .

Theorem 12. If is a bipolar fuzzy implicative ideal of , then for all ,(1)(2)

Proof. Let such that . Using Proposition 2, we have and . It follows thatNow, let such that , that is,Since is a bipolar fuzzy implicative ideal of , it follows from Theorem 7 and Proposition 2 thatThis completes the proof.

Theorem 13. Let be a bipolar fuzzy set in satisfying Theorem 12 (1). Then, is a bipolar fuzzy implicative ideal of .

Proof. We first prove that is a bipolar fuzzy ideal of . Let such that . Then,which implies from (b1) and Theorem 12 (1) that and . Therefore, by Proposition 2, we know that is a bipolar fuzzy ideal of . Note that for all . Using Theorem 12 (1) and Definition 1 (i), we haveand so, is a bipolar fuzzy implicative ideal of by Theorem 6.

Theorem 14. Let be a bipolar fuzzy set in satisfying Theorem 12 (2). Then, is a bipolar fuzzy implicative ideal of .

Proof. Let such that , that is,Then,and so, is a bipolar fuzzy implicative ideal of by Theorem 13.

Corollary 1. If is a bipolar fuzzy implicative ideal of , thenwhenever for all .

Proof. Let such that . Then,This completes the proof.

Theorem 15. (Extension Property). Let and be bipolar fuzzy ideals of such that and and and for all . If is a bipolar fuzzy implicative ideal of , then so is .

Proof. We assume that is a bipolar fuzzy implicative ideal of . For any , we haveIt follows from Definition 1 (i) and (ii) thatfor all . Hence, by Theorem 7, is a bipolar fuzzy implicative ideal of .

5. Conclusions

In the present paper, we apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain more related results. We considered the relation of a bipolar fuzzy ideal with a bipolar fuzzy implicative ideal and provided characterizations of a bipolar fuzzy implicative ideal. Also, we studied conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal. Furthermore, an extension property for a bipolar fuzzy implicative ideal is discussed.

We hope that this work will give a deep impact on the upcoming research in this field and other fuzzy algebraic study to open up new horizons of interest and innovations. One may apply this concept to study some application fields such as decision making, knowledge base system, and data analysis. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras, B-algebras, MV-algebras, d-algebras, and Q-algebras.

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 work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

References

W. R. Zhang, “Bipolar fuzzy sets and relations: a computational framework forcognitive modeling and multiagent decision analysis,” in Proceedings of the First International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference, pp. 305–309, San Antonio, TX, USA, December 1994.
View at: Google Scholar
K. M. Lee, “Bipolar-valued fuzzy sets and their operations,” in Proceedings of the International Conference on Intelligent Technologies, pp. 307–312, Bangkok, Thailand, 2000.
View at: Google Scholar
K. M. Lee, “Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets,” International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 2, pp. 125–129, 2004.
View at: Publisher Site | Google Scholar
W. R. Zhang and Y. Yang, “Bipolar fuzzy sets,” in Proceedings of the 1998 IEEE International Conference on Fuzzy Systems, pp. 835–840, Anchorage, AK, USA, May 1998.
View at: Google Scholar
K. J. Lee, “Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 32, pp. 361–373, 2009.
View at: Google Scholar
K. J. Lee and Y. B. Jun, “Bipolar fuzzy a-ideals of BCI-algebras,” Communications of the Korean Mathematical Society, vol. 26, no. 4, pp. 531–542, 2011.
View at: Publisher Site | Google Scholar
Y. B. Jun, M. S. Kang, and H. S. Kim, “Bipolar fuzzy implicative hyper BCK-ideals in hyper BCK-algebras,” vol. 69, no. 2, pp. 175–186, 2009, http://www.jams.or.jp/notice/scmj/smj.html.
View at: Google Scholar
Y. B. Jun, M. S. Kang, and H. S. Kim, “Bipolar fuzzy structures of some types of ideals in hyper BCK-algebras,” vol. 70, no. 1, pp. 109–121, 2009, http://www.jams.or.jp/notice/scmj/smj.html.
View at: Google Scholar
Y. B. Jun, M. S. Kang, and H. S. Kim, “Bipolar fuzzy hyper BCK-ideals in hyper BCK-algebras,” Iranian Journal of Fuzzy Systems, vol. 8, no. 2, pp. 105–120, 2011.
View at: Google Scholar
Y. B. Jun, M. S. Kang, and S. Z. Song, “Several types of bipolar fuzzy hyper BCK-ideals in hyper BCK-algebras,” Honam Mathematical Journal, vol. 34, no. 2, pp. 145–159, 2012.
View at: Publisher Site | Google Scholar
S. Abdullah and M. M. M. Aslam, “Bipolar fuzzy ideals in LA-semigroups,” World Applied Sciences Journal, vol. 17, no. 12, pp. 1769–1782, 2012.
View at: Google Scholar
M. Akram and N. O. Alshehri, “Bipolar fuzzy Lie ideals,” Utilitas Mathematica, vol. 87, pp. 265–278, 2012.
View at: Google Scholar
M. Akram, W. Chen, and Y. Lin, “Bipolar fuzzy Lie superalgebras,” Quasigroups Related Systems, vol. 20, pp. 139–156, 2012.
View at: Google Scholar
M. Akram, A. B. Saeid, K. P. Shum, and B. L. Meng, “Bipolar fuzzy K-algebras,” International Journal of Fuzzy Systems, vol. 12, no. 3, pp. 252–258, 2010.
View at: Google Scholar
Y. B. Jun, K. J. Lee, and E. H. Roh, “Ideals and filters in CI-algebras based on bipolar-valued fuzzy sets,” Annals of Fuzzy Mathematics and Informatics, vol. 4, pp. 109–121, 2012.
View at: Google Scholar
Y. B. Jun and C. H. Park, “Filters of BCH-algebras based on bipolar-valued fuzzy sets,” International Mathematical Forum, vol. 4, no. 13, pp. 631–643, 2009.
View at: Google Scholar
Y. B. Jun and S. Z. Song, “Subalgebras and closed ideals of BCH-algebras based on bipolar-valued fuzzy sets,” vol. 68, no. 2, pp. 287–297, 2008, http://www.jams.or.jp/notice/scmj/smj.html.
View at: Google Scholar
K. Kawila, C. Udomsetchai, and A. Iampan, “Bipolar fuzzy UP-algebras,” Mathematical and Computational Applications, vol. 23, no. 4, p. 69, 2018.
View at: Publisher Site | Google Scholar
C. S. Kim, J. G. Kang, and J. M. Kang, “Ideal theory of semigroups based on the bipolar valued fuzzy sets theory,” Annals of Fuzzy Mathematics and Informatics, vol. 2, no. 2, pp. 193–206, 2011.
View at: Google Scholar
S. K. Majumder, “Bipolar valued fuzzy sets in -semigroups,” Mathematica Aeterna, vol. 2, no. 3, pp. 203–213, 2012.
View at: Google Scholar
G. Muhiuddin, “Bipolar fuzzy KU-subalgebras/ideals of KU-algebras,” Annals of Fuzzy Mathematics and Informatics, vol. 8, no. 3, pp. 409–418, 2014.
View at: Google Scholar
G. Muhiuddin, H. Harizavi, and Y. B. Jun, “Bipolar-valued fuzzy soft hyper BCK ideals in hyper BCK algebras,” Discrete Mathematics Algorithms and Applications, vol. 12, no. 2, Article ID 2050018, p. 16, 2020.
View at: Publisher Site | Google Scholar
S. Sabarinathan, P. Muralikrishna, and D. C. Kumar, “Bipolar valued fuzzy H-ideals of BF-algebras,” International Journal of Pure and Applied Mathematics, vol. 112, no. 5, pp. 87–92, 2017.
View at: Google Scholar
T. Senapati, “Bipolar fuzzy structure of BG-subalgebras,” The Journal of Fuzzy Mathematics, vol. 23, pp. 209–220, 2015.
View at: Google Scholar
T. Senapati, “On bipolar fuzzy B-subalgebras of B-algebras,” in Emerging Research on Applied Fuzzy Sets and Intuitionistic Fuzzy Matrices, IGI Publishing, Hershey, PA, USA, 2016.
View at: Google Scholar
N. Yaqoob and A. Saeid, “Some results in bipolar-valued fuzzy ordered AG-groupoids,” Discussiones Mathematicae-General Algebra and Applications, vol. 32, no. 1, pp. 55–76, 2012.
View at: Google Scholar
A. Al-Masarwah and A. G. Ahmad, “-Polar fuzzy ideals of BCK/BCI-algebras,” Journal of King Saud University-Science, vol. 31, no. 4, pp. 1220–1226, 2019.
View at: Publisher Site | Google Scholar
A. Al-Masarwah and A. G. Ahmad, “-Polar ( α, β ) -fuzzy ideals in BCK/BCI-algebras,” Symmetry, vol. 11, no. 1, p. 44, 2019.
View at: Publisher Site | Google Scholar
A. Al-Masarwah and A. G. Ahmad, “A new form of generalized -PF ideals in BCK/BCI-algebras,” Annals of Communications in Mathematics, vol. 2, no. 1, pp. 11–16, 2019.
View at: Google Scholar
A. Al-Masarwah, A. G. Ahmad, and A. Ghafur Ahmad, “On(complete) normality of m-pF subalgebras in BCK/BCI-algebras,” AIMS Mathematics, vol. 4, no. 3, pp. 740–750, 2019.
View at: Publisher Site | Google Scholar
G. Muhiuddin, M. M. Takallo, R. A. Borzooei, and Y. B. Jun, “m-polar fuzzy q-ideals in BCI-algebras,” Journal of King Saud University- Science, vol. 32, no. 6, pp. 2803–2809, 2020.
View at: Publisher Site | Google Scholar
D. Al-Kadi and G. Muhiuddin, “Bipolar fuzzy BCI-implicative ideals of BCI-algebras,” Annals of Communications in Mathematics, vol. 3, no. 1, pp. 88–96, 2020.
View at: Google Scholar
G. Muhiuddin, D. Al-Kadi, A. Mahboob, and K. P. Shum, “New types of bipolar fuzzy ideals of BCK-algebras,” International Journal of Analysis and Applications, vol. 18, no. 5, pp. 859–875, 2020.
View at: Google Scholar
C. Jana, T. Senapati, K. P. Shum, and M. Pal, “Bipolar fuzzy soft subalgebras and ideals of BCK/ BCI-algebras based on bipolar fuzzy points,” Journal of Intelligent and Fuzzy Systems, vol. 37, no. 2, pp. 2785–2795, 2019.
View at: Publisher Site | Google Scholar
C. Jana, T. Senapati, M. Pal et al., “Different types of cubic ideals in BCI-algebras based on fuzzy points,” vol. 31, pp. 367–381, 2020, https://www.springer.com/journal/13370.
View at: Publisher Site | Google Scholar
C. Jana and M. Pal, “Generalized intuitionistic fuzzy ideals of BCK/BCI-algebras based on 3-valued logic and its computational study,” Fuzzy Information and Engineering, vol. 9, no. 4, pp. 455–478, 2017.
View at: Publisher Site | Google Scholar
C. Jana and K. Shum, “Lukaswize triple-valued intuitionistic fuzzy BCK/BCI-subalgebras,” in Handbook of Research on Emerging Applications of Fuzzy Algebraic Structures, pp. 191–212, IGI Global, Philadelphia, PA, USA, 2020.
View at: Google Scholar
G. Muhiuddin, S. J. Kim, and Y. B. Jun, “Implicative N-ideals of BCK-algebras based on neutrosophic N-structures,” Discrete Mathematics Algorithms and Applications, vol. 11, no. 1, Article ID 1950011, 2019.
View at: Publisher Site | Google Scholar
G. Muhiuddin and S. Aldhafeeri, “N-Soft p-ideal of BCI-algebras,” European Journal of Pure and Applied Mathematics, vol. 12, no. 1, pp. 79–87, 2019.
View at: Publisher Site | Google Scholar
G. Muhiuddin and A. M. Al-roqi, “Classifications of (alpha, beta)-fuzzy ideals in -algebras,” Journal of Mathematical Analysis and Applications, vol. 7, no. 6, pp. 75–82, 2016.
View at: Google Scholar
G. Muhiuddin and S. Aldhafeeri, “Join hesitant fuzzy filters of residuated lattices,” Italian Journal of Pure and Applied Mathematics, vol. 43, pp. 100–114, 2020.
View at: Google Scholar
G. Muhiuddin, A. M. Alanazi, M. E. A. Elnair, and K. P. Shum, “Inf-hesitant fuzzy subalgebras and ideals in BCK/BCI-algebras,” European Journal of Pure and Applied Mathematics, vol. 13, no. 1, pp. 9–18, 2020.
View at: Publisher Site | Google Scholar
G. Muhiuddin and B. Young, “Jun, Sup-hesitant fuzzy subalgebras and its translations and extensions,” Annals of Communications in Mathematics, vol. 2, no. 1, pp. 48–56, 2019.
View at: Google Scholar
G. Muhiuddin and A. M. Al-roqi, “Regular hesitant fuzzy filters and MV -hesitant fuzzy filters of residuated lattices,” Journal of Computational Analysis and Applications, vol. 24, no. 6, pp. 1133–1144, 2018.
View at: Google Scholar
G. Muhiuddin and S. Aldhafeeri, “Subalgebras and ideals in BCK/BCI-algebras based on uni-hesitant fuzzy set theory,” European Journal of Pure and Applied Mathematics, vol. 11, no. 2, pp. 417–430, 2018.
View at: Publisher Site | Google Scholar
G. Muhiuddin, E. H. Roh, and S. Sun Shin AhnY. B. Jun, “Hesitant fuzzy filters in lattice implication algebras,” Journal of Computational Analysis and Applications, vol. 22, no. 6, pp. 1105–1113, 2017.
View at: Google Scholar
G. Muhiuddin, H. S. Kim, S. Z. Song, and Y. B. Jun, “Hesitant fuzzy translations and extensions of subalgebras and ideals in BCK/BCI-algebras,” Journal of Intelligent and Fuzzy Systems, vol. 32, no. 1, pp. 43–48, 2017.
View at: Publisher Site | Google Scholar
G. Muhiuddin, “Hesitant fuzzy filters and hesitant fuzzy -filters in residuated lattices,” Journal of Computational Analysis and Applications, vol. 20, no. 2, pp. 394–404, 2016.
View at: Google Scholar
G. Muhiuddin, N. Rehman, and Y. B. Jun, “A generalization of -fuzzy ideals in ternary semigroups,” Annals of Communications in Mathematics, vol. 2, no. 2, pp. 73–83, 2019.
View at: Google Scholar
G. Muhiuddin, “Neutrosophic subsemigroups,” Annals of Communication in Mathematics, vol. 1, no. 1, pp. 1–10, 2018.
View at: Google Scholar
G. Muhiuddin, “Cubic interior ideals in semigroups,” Applications and Applied Mathematics, vol. 14, no. 1, pp. 463–474, 2019.
View at: Google Scholar
G. Muhiuddin, A. Mahboob, and N. M. Khan, “A new type of fuzzy semiprime subsets in ordered semigroups,” Journal of Intelligent and Fuzzy Systems, vol. 37, no. 3, pp. 4195–4204, 2019.
View at: Publisher Site | Google Scholar
G. Muhiuddin and K. P. Shum, “New types of - fuzzy subalgebras of BCK/BCI-algebras,” International Journal of Mathematics and Computer Science, vol. 14, no. 2, pp. 449–464, 2019.
View at: Google Scholar

Copyright

Copyright © 2021 G. Muhiuddin and D. Al-Kadi. 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.

PDF Download Citation

Download other formats

Order printed copies