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(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

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


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:

(μA1(x),μA2(x),,μAp(x)):μA1(x)μA2(x)μAp(x).

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

μGi(g1g2)γ(μGi(g1)μGi(g2))α(1)

and

μGi((g3)1)γμGi(g3)α(2)

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

μGi(1)γμGi(g1)α(3)

g1G, where 1 is the identity of G.

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

μGi(1)γ=μGi((g1)1g1)γ(μGi((g1)1g1)γ)γ((μGi(g1)μGi(g1)1)α)γ=(μGi(g1)γ)(μGi(g11)γ)(αγ)μGi(g1)(μGi(g1)α)α=μGi(g1)α(4)

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

GμGi((g1)1g2)γ(μGi(g1)μGi(g2))α,g1,g2G.(5)

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

μGi((g1)1g2)γ=μGi((g1)1g2)γγ((μGi(g2)μGi(g1)1)α)γ=(μGi(g2)μGi(g11)γ)(αγ)μGi(g2)(μGi(g1)α)α=(μGi(g1)μGi(g2))α.(6)

Conversely, assume

μGi((g1)1g2)γ(μGi(g1)μGi(g2))α,g1,g2G,(7)

then

μGi(1)γ=μGi((g1)1g2)γ(μGi(g1)μGi(g1))α=μGi(g1)α.(8)

So

μGi((g1)1)γ=μGi((g1)11)γ=μGi((g1)11)γγ=(μGi(g1)μGi(1)α)γμGi(g2)(μGi(g1)α)α=(μGi(g1)μGi(g2))α.(9)

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

μGi((g1)1g2)γμGi(g1)μGi(g2)α,g1,g2G.(10)

If there exists g3,g4G such that

μGi((g3)1g4)γ=δ>μGi(g3)μGi(g4)α,(11)

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

μGi((g1)1g2)γμGi(g1)μGi(g2)α,(12)

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

(ii)(i)

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

μGi((g1)1g2)γμGi(g1)μGi(g2)α<δδα=δα=α.(13)

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

μG1i×μG2i(g1,g2)=μG1i(g1)μG2i(g2),(g1,g2)G1×G2.(14)

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

μG1i(g111)γμG1i(g11)α(15)

and

μG2i(g211)γμG2i(g21)α.(16)

(g12,g22)G1×G2. We have

((μG1i×μG2i)(g11,g21)1,(g12,g22))γ=(μG1i×μG2i)((g111,g211),(g12,g22))γ=(μG1i(g111,g12)μG2i(g211,g22))γ=(μG1i(g111,g12)γ)(μG2i(g211,g22)γ)(μG1i(g11)μG1i(g12)α)(μG2i(g21)μG2i(g22)α)(μG1i(g11)μG1i(g12)α)(μG2i(g21)μG2i(g22)α)=(μG1i(g11)μG2i(g21))(μG1i(g12)μG2i(g22))α=(μG1i×μG2i(g11,g21))(μG1i×μG2i(g12,g22))α.(17)

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.

μG1i(11)γμG2i(a)α,aG2(18)

and

μG2i(12)γμG1i(g1)α,g1G1.(19)

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

μG1i×μG2i(g1,a)α=(μG1i(g1)μG2i(a))α,=(μG1i(g1)α)(μG2i(a)α)<(μG1i(11)γ)(μG2i(12)γ)=μG1i×μG2i(11,12)γ.(20)

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

μG1i×μG2i(g1,a)α<μG1i×μG2i(11,12)γ.(21)

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

f(μG1i)(g2)=infg1G1{μG1i(g1):f(g1)=g2},g2G2(22)

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

(f(μG1i)((g21)1g22))γ1=(f(μG1i)(g21)f(μG1i)(g22))α.(23)

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

(f(μG1i)((g21)1g22))γ=infkG1{μG1i(k):f(k)=(g21)1g22}γ=infkG1{μG1i(k)γ:f(k)=(g21)1g22}infg11,g12G1{μG1i((g11)1g12)γ:f(g11)=g21,f(g12)=g22}infg11,g12G1{(μG1i)(g11)f(μG1i)(g12)α:f(g11)=g21,f(g12)=g22}=(infg11G1{(μG1i)(g11):f(g11)=g21}infg12G1{(μG1i)(g12):f(g12)=g22})α=(f(μG1i)(g21)f(μG1i)(g22))α(24)

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

f1(μG2i)(g1)=μG2i(f(g1)),g1G1.(25)

Proof For any g11,g12G1

f1(μG2i)((g11)1g12)γ=μG2i(f((g11)1g12))γ=μG2i(f((g11)1)f(g12))γ(μG2i(f(g11))μG2i(f(g12)))α=(f1(μG2i)(g11)f1(μG2i)(g12))α.

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

μGi(g1g2g11)γμGi(g2)α(26)

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,

μGi(g1g2)γμGi(g2g1)α.(27)

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

μGi([g1,g2])γμGi(g1)α(28)

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

Proof For any g1,g2G,

μGi([g1,g2])γ=μGi(g11g21g1g2)γ=μGi(g11(g21g1g2))γγ(μGi(g11)μGi(g21g1g2)α)γ=(μGi(g11)γ)(μGi(g21g1g2)γ)α(29)

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,

μGi([g1,g2])γμGi(g1)α(30)

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

μGi([g1,g2])γ=μGi(g1g11g21g1g2)γ=μGi(g1(g11g21g1g2))γγ(μGi(g1)μGi(g11g21g1g2)α)γ=(μGi(g1)γ)(μGi(g11g21g1g2)γ)α(μGi(g1)γ)(μGi(g1)α)αμGi(g1)μGi(g1)α=μGi(g1)α(31)

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

μGi([g1,g2])γ=μGi(e)γμGi(g1)α(32)

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.

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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


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