iconOpen Access



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

Memet Şahin1, Vakkas Uluçay2, S. A. Edalatpanah3,*, Fayza Abdel Aziz Elsebaee4, Hamiden Abd El-Wahed Khalifa5

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: email

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


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.


1  Introduction

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 [68], on fuzzy multi sets [911] on anti-fuzzy group theory [1217] 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.

2  Preliminary

In this paper, G,G1 and G2 stands for groups with identities 1,11 and 12, respectively. In the rest of the article, we will always suppose that 0α<γ1.

Definition 2.1 [3] Let A be a fuzzy subset of G. A is called a fuzzy subgroup of G if, for all x.yG,

     i)  A(xy)A(x)A(y),

    ii)  A(x1)A(x).

Definition 2.2 [9] Let A be a fuzzy subset of G. A is called a (α,γ)-anti-fuzzy subgroup of G if, for all x.y,zG,

     i)  A(xy)γ(A(x)A(y))α,

    ii)  A(z1)αA(z)γ.

Definition 2.3 [10] Let E be a non-empty set and Q be the set of all crisp multisets drawn from the interval [0,1]. A fuzzy multiset A drawn from E is represented by a function CMA:EQ.

The value CMA(x), mentioned above, is a crisp multiset drawn from [0,1]. For each xE, CMA(x), is defined as the decreasingly ordered sequence of elements and it is denoted by:


A fuzzy set on a set E can be understood as a special case of fuzzy multiset where CMA(x)=μA1(x) for all xE.

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

Definition 3.1 A fuzzy set A of a group G is called a (α,γ)-anti fuzzy multi subgroup of G if g1,g2,g3G




where (g3)1 is the inverse element of (g3).

Proposition 3.2 If A is a (α,γ)-anti-fuzzy-multi-subgroup of a group G, then


g1G, where 1 is the identity of G.

Proof g1G and let (g1)1 be the inverse element of (g1). Then


Theorem 3.3 Let A be multi fuzzy subset of a group G. Then A is a (α,γ)-anti-fuzzy multi subgroup of


Proof Let A is a (α,γ)-anti-fuzzy multi group of G, then


Conversely, assume






In this way A is a (α,γ)-anti-fuzzy multi-subgroup of G.

Theorem 3.4 Let A be a fuzzy multi subset of a group G. Then the following are equivalent:

     i)  A(δ) is a multi-subgroup of G, δ(α,γ], where A(δ);

    ii)  A is a (α,γ)-anti-fuzzy multi subgroup of G.

Proof (i)(ii) let A(δ) is a multi-subgroup of G. We need to prove that


If there exists g3,g4G such that


Then μGi(g3)<δ,μGi(g4)<δ and δ(α,γ]. Thus μGi(g3)A(δ),μGi(g4)A(δ). But μGi((g3)1g4)δ, that is (g3)1g4A(δ). This is a contradiction with that A(δ) is a multi-subgroup of G. Hence


Holds g1,g2G. Therefore, A is a (α,γ)-anti-fuzzy multi subgroup of G.


Let A is a (α,γ)-anti-fuzzy multi subgroup of G. δ(α,γ], such that A(δ), we need to show that (g1)1g2A(δ), g1,g2A(δ). Since μGi(g1)<δ,μGi(g2)<δ then


Note that δ<γ, we have μGi((g1)1g2)<δ. Thus (g1)1g2A(δ). We set inf=1, where is the empty set.

Theorem 3.5 Let A and B are two fuzzy multi-subsets of groups G1 and G2, respectively. The product of A and B, denoted by μG1i×μG2is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2, where


Proof Let g211 is the inverse element of g21 in G2 and g111 is the inverse element of g11 in G1. Then(g111,g211) be the inverse element of (g11,g21)G1×G2. Hence




(g12,g22)G1×G2. We have


Hence μG1i×μG2i s a (α,γ)-anti-fuzzy multi-subgroup of G1×G2.

Theorem 3.6 Let A and B are two fuzzy multi-subsets of groups G1 and G2, respectively. If A×B is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2, then at least one of the following statements must hold.




Proof Let A×B is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2. By contraposition, assume that none of the statements holds. Then we can find g1G1 and aG2 such that μG1i(g1)α<μG2i(12)γ and μG2i(a)α<μG1i(11)γ. Now


Therefore A×B is a (α,γ)-anti-fuzzy multi-subgroup of G1×G2 satisfying


This is a contradict with that (11,12) is the identity of G1×G2.

Theorem 3.7 Let f:G1G2 is a homomorphism and let A is a (α,γ)-anti-fuzzy multi-subgroup of G1. Then f(μG1i) is a (α,γ)-anti-fuzzy multi-subgroup of G2, where


Proof If f1(g21)= or f1(g22)= for any g21,g22G2, then


Assume that f1(g21)= or f1(g22)= for any g21,g22G2 then


Thus f(μG1i) is a (α,γ)-anti-fuzzy multi-subgroup of G2.

Theorem 3.8 Let f:G1G2 is a homomorphism and let A is a (α,γ)-anti-fuzzy multi-subgroup of G2. Then f1(μG2i) is a (α,γ)-anti-fuzzy multi-subgroup of G1, where


Proof For any g11,g12G1


Thus f1(μG2i) is a (α,γ)-anti-fuzzy multi-subgroup of G1.

Definition 3.9 Let A is a (α,γ)-anti fuzzy multi subgroup of G. A is called a (α,γ)-anti fuzzy normal subgroup of G if, g1,g2G


Proposition 3.10 Let A is a (α,γ)-anti fuzzy multi subgroup of G. A is a (α,γ)-anti fuzzy normal subgroup if and only if, g1,g2G,


Proposition 3.11 Let A is a (α,γ)-anti fuzzy multi subset of G. Then A is a (α,γ)-anti fuzzy multi normal subgroup of G if and only if Aδ is a normal subgroup of G δ(α,γ].

Proposition 3.12 Let A is a (α,γ)-anti fuzzy multi normal subgroup of G and g1G.

     i)  If μGi(g1)α, then μGi(g1g2g11)α for all g2G.

    ii)  If γ<μGi(g1)<α, then μGi(g2g1g21)=μGi(g1) for all g2G.

   iii)  If g2G and γ<μGi(g1g2)<α, then μGi(g1g2)=μGi(g2g1).

    iv)  If g2G and μGi(g1g2)α, then, μGi(g2g1)α

     v)  If g2G and μGi(g1g2)γ, then, μGi(g2g1)γ.

Proof (i) μGi(g1)α, then g1Aα. By proposition 9, Aα is a fuzzy multi normal subgroup of G and thus g1g2g11Aα. Hence μGi(g1g2g11)α. (ii) Let μGi(g1)=δ. Then γ<δ<α. By proposition 9, Aδ is a fuzzy multi normal subgroup of G. Hence g1g2g11Aδ; that is μGi(g1g2g11)δ=μGi(g1). Suppose μGi(g1g2g11)<δ. Set δ0=min{μGi(g1g2g11),α}. Then γ<δ0<α. By proposition 9, Aδ0 is a fuzzy multi normal subgroup of G, and thus g1g2g11Aδ0. Therefore g1=g21(g1g2g11)g2Aδ0; that is μGi(g1g2g11)δ0<δ which is a contradiction to that μGi(g1)=δ. As result, μGi(g2g1g21)=μGi(g1).

(iii) If γ<μGi(g1g2)<α, then μGi(g2g1)=μGi(g11(g2g1)g1)=μGi(g1g2) by (ii); that is μGi(g1g2)=μGi(g2g1).

(iv) μGi(g1g2)α, then g1g2Aα. Since Aα is a fuzzy multi normal subgroup of G by Proposition 9, (g2g1)=g11(g1g2)g1Aα; that is μGi(g2g1)α.

(v) Assume that μGi(g2g1)<γ on the contrary. If μGi(g2g1)α, then, by (i) μGi(g1g2)α, which is contradictory to that μGi(g1g2)>γ. If μGi(g2g1)<γ, then, by (iii), μGi(g1g2)=μGi(g2g1)<α, which is contradictory to that μGi(g1g2)γ. Therefore μGi(g2g1)γ.

Proposition 3.13 Let A is a (α,γ)-anti fuzzy multi subgroup of G. Then A is a (α,γ)-anti fuzzy multi normal subgroup of G if and only if


for all g1,g2G, where [g1,g2]=g11g21g1g2 is a commutator in G.

Proof For any g1,g2G,


Since A is a (α,γ)-anti fuzzy multi normal subgroup of G, μGi(g11)γμGi(g1)α and μGi(g21g1g2)γμGi(g1)α. Hence,


Conversely, if μGi([g1,g2])γμGi(g1)α, then


Therefore A is a (α,γ)-anti fuzzy multi normal subgroup of G.

Proposition 3.14 If G is an abelian group and A is a (α,γ)-anti fuzzy multi subgroup of G, then A is a (α,γ)-anti fuzzy multi normal subgroup of G.

Proof Since G is an abelian group, we have [g1g2]=e; hence


for all g1,g2G. By Proposition 11, A is a (α,γ)-anti fuzzy multi normal subgroup of G.

4  Conclusions

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.


 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
Şahin, M., Uluçay, V., Edalatpanah, S.A., Elsebaee, F.A.A., Khalifa, H.A.E. (2023). (<b>α</b>, <b>γ</b>)-anti-multi-fuzzy subgroups and some of its properties. Computers, Materials & Continua, 74(2), 3221-3229. https://doi.org/10.32604/cmc.2023.033006
Vancouver Style
Şahin M, Uluçay V, Edalatpanah SA, Elsebaee FAA, Khalifa HAE. (<b>α</b>, <b>γ</b>)-anti-multi-fuzzy subgroups and some of its properties. Comput Mater Contin. 2023;74(2):3221-3229 https://doi.org/10.32604/cmc.2023.033006
IEEE Style
M. Şahin, V. Uluçay, S.A. Edalatpanah, F.A.A. Elsebaee, and H.A.E. Khalifa "(<b>α</b>, <b>γ</b>)-Anti-Multi-Fuzzy Subgroups and Some of Its Properties," Comput. Mater. Contin., vol. 74, no. 2, pp. 3221-3229. 2023. https://doi.org/10.32604/cmc.2023.033006

cc This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • 841


  • 543


  • 0


Share Link