Open Access

ARTICLE

# (**α**, **γ**)-Anti-Multi-Fuzzy Subgroups and Some of Its Properties

1 Department of Mathematics, Gaziantep University, Gaziantep, Turkey

2 Department of Mathematics, Kilis 7 Aralık University, Kilis, Turkey

3 Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran

4 Department of Mathematics, Helwan University, Cairo, Egypt

5 Department of Mathematics, Qassim University, Alasyah, Saudi Arabia

* Corresponding Author: S. A. Edalatpanah. Email:

*Computers, Materials & Continua* **2023**, *74*(2), 3221-3229. https://doi.org/10.32604/cmc.2023.033006

**Received** 04 June 2022; **Accepted** 09 August 2022; **Issue published** 31 October 2022

## Abstract

Recently, fuzzy multi-sets have come to the forefront of scientists’ interest and have been used in algebraic structures such as multi-groups, multi-rings, anti-fuzzy multigroup and (α, γ)-anti-fuzzy subgroups. In this paper, we first summarize the knowledge about the algebraic structure of fuzzy multi-sets such as (α, γ)-anti-multi-fuzzy subgroups. In a way, the notion of anti-fuzzy multigroup is an application of anti-fuzzy multi sets to the theory of group. The concept of anti-fuzzy multigroup is a complement of an algebraic structure of a fuzzy multi set that generalizes both the theories of classical group and fuzzy group. The aim of this paper is to highlight the connection between fuzzy multi-sets and algebraic structures from an anti-fuzzification point of view. Therefore, in this paper, we define (α, γ)-anti-multi-fuzzy subgroups, (α, γ)-anti-multi-fuzzy normal subgroups, (α, γ)-anti-multi-fuzzy homomorphism on (α, γ)-anti-multi-fuzzy subgroups and these been explicated some algebraic structures. Then, we introduce the concept (α, γ)-anti-multi-fuzzy subgroups and (α, γ)-anti-multi-fuzzy normal subgroups and of their properties. This new concept of homomorphism as a bridge among set theory, fuzzy set theory, anti-fuzzy multi sets theory and group theory and also shows the effect of anti-fuzzy multi sets on a group structure. Certain results that discuss the (α, γ) cuts of anti-fuzzy multigroup are explored.## Keywords

Dresher et al. [1] laid the foundations of the theory of multigroup in 1938. Zadeh [2] introduced the concept of a fuzzy subset of a set, fuzzy set are a kind of useful mathematical structure to represent a collection of objects whose boundary is uncertainty in 1965. Therefore, on the basis of fuzzy set theory, Sebastian et al. [3] introduced Multi-Fuzzy Sets, Atanassov [4] proposed intuitionistic fuzzy set theory, Shinoj et al. [5] initiated intuitionistic fuzzy multisets. Recently, the above theories have developed in many directions and found its applications in a wide variety of fields including algebraic structures. For example, on fuzzy sets [6–8], on fuzzy multi sets [9–11] on anti-fuzzy group theory [12–17] are some of the selected works. Rosenfeld [18] defined the notion of fuzzy subgroup. Biswas [19] introduced the concept of anti-fuzzy subgroup of group. Yuan et al. [20] introduced the concept of fuzzy subgroup with thresholds. A fuzzy subgroup with thresholds lambda and mu is also called a (lambda, mu)-fuzzy subgroup. Yao [21] defined (lambda, mu)-fuzzy normal subgroups and (lambda, mu)-fuzzy quotient subgroups these examined some properties. On these studies, Shen [22] defined anti-fuzzy subgroups and Dong [23] introduced the product of anti-fuzzy subgroups. Then, Feng et al. [24] introduces the notion of (lambda, mu)-anti-fuzzy subgroups and discussed some properties. Since the idea of anti-multi fuzzy subgroup has been extended to multi fuzzy subgroups, it is expedient to explore the idea in (α, γ)-anti-multi-fuzzy subgroups setting. The motivation of this paper is to extend the notions of anti-multi fuzzy subgroups and (α, γ)-anti-multi-fuzzy subgroups to fuzzy multigroup environment and to present some new results. Moreover, this research proposes the generalization of the results known for (α, γ)-anti-multi-fuzzy subgroups. It is known that the notion of fuzzy multiset is well entrenched in solving many real-life problems. So, the algebraic structure defined concerning them in this paper could help to approach these issues from a different position. The benefit of this paper is the link found between algebraic structures and fuzzy multisets by introducing (α, γ)-anti-multi-fuzzy subgroups and studying their properties.

The outlines are presented as follows: Section 2 presents some foundational notions relevant to the study, whereas the main results are reported in Section 3. In Section 4, we make some concluding remarks and suggestions for future work.

In this paper,

Definition 2.1 [3] Let

i)

ii)

Definition 2.2 [9] Let

i)

ii)

Definition 2.3 [10] Let

The value

A fuzzy set on a set

3 (α, γ)-Anti-Multi Fuzzy Subgroups and Some of Its Properties

Definition 3.1 A fuzzy set

and

where

Proposition 3.2 If

Proof

Theorem 3.3 Let

Proof Let

Conversely, assume

then

So

In this way

Theorem 3.4 Let

i)

ii)

Proof

If there exists

Then

Holds

Let

Note that

Theorem 3.5 Let

Proof Let

and

Hence

Theorem 3.6 Let

and

Proof Let

Therefore

This is a contradict with that

Theorem 3.7 Let

Proof If

Assume that

Thus

Theorem 3.8 Let

Proof For any

Thus

Definition 3.9 Let

Proposition 3.10 Let

Proposition 3.11 Let

Proposition 3.12 Let

i) If

ii) If

iii) If

iv) If

v) If

Proof (i)

(iii) If

(iv)

(v) Assume that

Proposition 3.13 Let

for all

Proof For any

Since

Conversely, if

Therefore

Proposition 3.14 If

Proof Since

for all

The aim of this paper was to highlight the function between (α, γ)-anti-multi-fuzzy subgroups and algebraic structures from other a point of view. It is well known that the concept of fuzzy multi set is well established in dealing with many real-life problems. So, the algebraic structure defined concerning them in this paper would help to approach these problems with a different perspective.

In this paper, we have defined the notion of (α, γ)-anti-multi-fuzzy subgroups and this structure some algebraic properties were developed. In this article, we have discussed (α, γ)-anti-multi-fuzzy subgroups, (α, γ)-anti-multi-fuzzy normal subgroups and defined (α, γ)-anti-multi-fuzzy homomorphism on (α, γ)-anti-multi-fuzzy subgroups. Interestingly, it has been observed that (α, γ)-anti-multi-fuzzy concept adds another dimension to the defined anti-fuzzy multi normal subgroups. This concept can further be extended for new results.

Funding Statement: Yibin University Pre-research Project, Research on the coupling and coordinated development of manufacturing and logistics industry under the background of intelligent manufacturing, (2022YY001); Sichuan Provincial Department of Education Water Transport Economic Research Center, Research on the Development Path and Countermeasures of the Advanced Manufacturing Industry in the Sanjiang New District of Yibin under a “dual circulation” development pattern (SYJJ2020A06).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

## References

1. M. Dresher and O. Ore, “Theory of multigroup,” American Journal of Mathematics, vol. 60, no. 3, pp. 705–733, 1938. [Google Scholar]

2. L. A. Zadeh, “Fuzzy sets,” Information Control, vol. 8, no. 3, pp. 338–353, 1965. [Google Scholar]

3. S. Sebastian and T. T. Ramakrishnan, “Multi-fuzzy sets,” International Mathematical Forum, vol. 5, pp. 2471–2476, 2010. [Google Scholar]

4. K. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Set System, vol. 20, no. 1, pp. 87–96, 1986. [Google Scholar]

5. T. K. Shinoj and S. S. John, “Intuitionistic fuzzy multisets and its application in medical diagnosis,” World Academy of Science, Engineering and Technology, vol. 6, pp. 1418–1421, 2012. [Google Scholar]

6. S. Abdullah and M. A. Naeem, “New type of interval valued fuzzy normal subgroups of groups,” New Trends in Mathematical Sciences, vol. 3, pp. 62–77, 2015. [Google Scholar]

7. J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, “Fuzzy subsets and fuzzy subgroups,” in Fuzzy Group Theory, 1st ed., vol. 1. Berlin, Heidelberg, Germany: Springer, pp. 1–39, 2005. [Google Scholar]

8. Y. L. Liu, “Quotient groups induced by fuzzy subgroups,” Quasigroups Related System, vol. 11, pp. 71–78, 2004. [Google Scholar]

9. A. Baby, T. K. Shinoj and S. J. John, “On abelian fuzzy multi groups and orders of fuzzy multi groups,” Journal of New Theory, vol. 5, pp. 80–93, 2015. [Google Scholar]

10. R. Muthuraj and S. Balamurugan, “Multi-fuzzy group and its level subgroups,” Gen, vol. 17, pp. 74–81, 2013. [Google Scholar]

11. Y. Tella, “On algebraic properties of fuzzy membership sequenced multisets,” British Journal of Mathematics & Computer Science, vol. 6, no. 2, pp. 146–164, 2015. [Google Scholar]

12. P. A. Ejegwa, J. A. Awolola, J. M. Agbetayo and I. M. Adamu, “On the characterisation of anti-fuzzy multigroups,” Annals of Fuzzy Mathematics and Informatics, vol. 21, no. 3, pp. 307–318, 2021. [Google Scholar]

13. S. Hoskova-Mayerova and M. Al Tahan, “Anti-fuzzy multi-ideals of near ring,” Mathematics, vol. 9, no. 5, pp. 494–504, 2021. [Google Scholar]

14. S. Gayen, S. Jha, M. K. Singh and A. K. Prasad, “On anti-fuzzy subgroup,” Yugoslav Journal of Operations Research, vol. 31, no. 4, pp. 539–546, 2021. [Google Scholar]

15. R. Sumitha and S. Jayalakshmi, “Anti-fuzzy quasi-ideals of near subtraction semigroups,” Malaya Journal of Matematik, vol. S, no. 1, pp. 261–264, 2021. [Google Scholar]

16. K. R. Balasubramanian and R. Revathy, “Product of (λ, μ)-multifuzzy subgroups of a group,” Annals of the Romanian Society for Cell Biology, vol. 25, no. 6, pp. 2448–2460, 2021. [Google Scholar]

17. S. H. Asaad and A. S. Mohammed, “New properties of anti-fuzzy ideals of regular semigroups,” IbnAL-Haitham Journal for Pure and Applied Sciences, vol. 32, no. 3, pp. 109–116, 2019. [Google Scholar]

18. A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, no. 3, pp. 512–551, 1971. [Google Scholar]

19. R. Biswas, “Fuzzy subgroups and anti-fuzzy subgroups,” Fuzzy Sets and Systems, vol. 35, pp. 121–124, 1990. [Google Scholar]

20. X. Yuan, C. Zhang and Y. Ren, “Generalized fuzzy groups and many-valued implications,” Fuzzy Sets System, vol. 138, no. 1, pp. 205–211, 2003. [Google Scholar]

21. B. Yao, “(λ, μ)-fuzzy normal subgroups and (λ, μ)-fuzzy quotient subgroups,” Journal of Fuzzy Mathematics, vol. 13, no. 3, pp. 695–705, 2005. [Google Scholar]

22. Z. Shen, “The anti-fuzzy subgroup of a group,” Journal of Liaoning Normat. Univ. (Nat. Sci.), vol. 18, no. 2, pp. 99–101, 1995. [Google Scholar]

23. B. Dong, “Direct product of anti-fuzzy subgroups,” Journal of Shaoxing Teachers College, vol. 5, pp. 29–34, 1992. [Google Scholar]

24. Y. Feng and B. Yao, “On (λ, μ)-anti-fuzzy subgroups,” Journal of Inequalities and Applications, vol. 1, pp. 1–5, 2012. [Google Scholar]

## Cite This Article

**APA Style**

*Computers, Materials & Continua*,

*74*

*(2)*, 3221-3229. https://doi.org/10.32604/cmc.2023.033006

**Vancouver Style**

**IEEE Style**

*Comput. Mater. Contin.*, vol. 74, no. 2, pp. 3221-3229. 2023. https://doi.org/10.32604/cmc.2023.033006