# A Study of(λ,μ)-Fuzzy Subgroups

Authors: Yuying Li; Xuzhu Wang; Liqiong Yang
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

We deal with topics regarding(λ,μ)-fuzzy subgroups, mainly (λ,μ)-fuzzy cosets and (λ,μ)-fuzzy normal subgroups. We give basic properties of (λ,μ)-fuzzy subgroups and present some results related to (λ,μ)-fuzzy cosets and (λ,μ)-fuzzy normal subgroups.


---

## Body

## 1. Introduction

Since Zadeh [1] introduced the concept of a fuzzy set in 1965, various algebraic structures have been fuzzified. As a result, the theory of fuzzy group was developed. In 1971, Rosenfeld [2] introduced the notion of a fuzzy subgroup and thus initiated the study of fuzzy groups.In recent years, some variants and extensions of fuzzy groups emerged. In 1996, Bhakat and Das proposed the concept of an(∈,∈∨q)-fuzzy subgroup in [3] and investigated their fundamental properties. They showed that A is an (∈,∈∨q)-fuzzy subgroup if and only if Aα is a crisp group for any α∈(0,0.5] provided Aα≠∅. A question arises naturally: can we define a type of fuzzy subgroups such that all of their nonempty α-level sets are crisp subgroups for any α in an interval (λ,μ]? In 2003, Yuan et al. [4] answered this question by defining a so-called (λ,μ)-fuzzy subgroups, which is an extension of (∈,∈∨q)-fuzzy subgroup. As in the case of fuzzy group, some counterparts of classic concepts can be found for (λ,μ)-fuzzy subgroups. For instance, (λ,μ)-fuzzy normal subgroups and (λ,μ)-fuzzy quotient groups are defined and their elementary properties are investigated, and an equivalent characterization of (λ,μ)-fuzzy normal subgroups was presented in [5]. However, there is much more research on (λ,μ)-fuzzy subgroups if we consider rich results both in the classic group theory and the fuzzy group theory in the sense of Rosenfeld.In this paper, we conduct a detailed investigation on(λ,μ)-fuzzy subgroups. The research includes further properties of (λ,μ)-fuzzy subgroups, (λ,μ)-fuzzy normal subgroups, and (λ,μ)-fuzzy left and right cosets.The rest of this paper is organized as follows. In Section2, we give properties of (λ,μ)-fuzzy subgroups. In Section 3, we present properties of (λ,μ)-fuzzy normal subgroups. In Section 4, we define (λ,μ)-fuzzy left and right cosets and discuss their properties.


## 2. Properties of(λ,μ)-Fuzzy Subgroups

In this paper,G stands for a group with identity e, and 0⩽λ<μ⩽1. By a fuzzy subset of G, we mean a mapping from G to the closed unit interval [0,1]. In this section, we present some basic properties of (λ,μ)-fuzzy subgroups.Definition 1.
LetA be a fuzzy subset of G. A is called a fuzzy subgroup of G if, for all x,y∈G, (i)
A(xy)⩾A(x)∧A(y),(ii)
A(x-1)⩾A(x).Definition1 was introduced by Rosenfeld [2] in 1971.Definition 2 (see [4]).
LetA be a fuzzy subset of G. A is called a (λ,μ)-fuzzy subgroup of G if, for all x,y∈G, (i)
A(xy)∨λ⩾A(x)∧A(y)∧μ,(ii)
A(x-1)∨λ⩾A(x)∧μ.Clearly, a(0,1)-fuzzy subgroup is just a fuzzy subgroup, and thus a (λ,μ)-fuzzy subgroup is a generalization of fuzzy subgroup.Definition 3.
For a fuzzy subsetA of G and α∈[0,1], let Aα={x∈G∣A(x)⩾α}. Then Aα is called a level subset of A.Proposition 4 (see [2]).
A fuzzy subsetA of G is a fuzzy subgroup if and only if Aα≠∅ is a crisp subgroup of G for every α∈(0,1].Proposition 5 (see [5]).
IfA is a (λ,μ)-fuzzy subgroup of G, then A(e)∨λ⩾A(x)∧μ for all x∈G.Corollary 6.
LetA be a (λ,μ)-fuzzy subgroup of G. Then (i)
IfA(x)⩾μ for some x∈G, then A(e)⩾μ.(ii)
IfA(x)<μ for all x∈G and H={x∣λ<A(x)<μ}≠∅, then A(e)=max{A(x)|x∈G}.(iii)
IfA(e)⩽λ, then A(x)⩽λ for all x∈G.Proof.
(i) IfA(x)⩾μ for some x∈G, then A(e)∨λ⩾A(x)∧μ=μ by Proposition 5; that is, A(e)⩾μ.
(ii) Whenx∈H, by Proposition 5, A(e)∨λ⩾A(x)∧μ=A(x); that is, A(e)⩾A(x). Thus A(e)>λ.
Whenx∉H, A(x)⩽λ due to A(x)<μ. A(e)=A(e)∨λ⩾A(x)∧μ=A(x) by Proposition 5. Therefore, for all x∈G, A(x)≤A(e); that is, A(e)=max{A(x)∣x∈G}.
(iii) IfA(e)⩽λ, then λ=A(e)∨λ⩾A(x)∧μ for all x∈G, which implies A(x)⩽λ due to λ<μ.Corollary 7.
LetA be a (λ,μ)-fuzzy subgroup of G and λ<A(e)<μ. Then A(x)⩽A(e) holds for all x∈G.Proof.
IfA(x)⩾μ for some x∈G, then A(e)⩾μ by Corollary 6(i), which is a contradiction. Hence A(x)<μ for all x∈G. Since e∈H, A(x)⩽A(e) holds for all x∈G by Corollary 6(ii).Corollary 8.
LetA be a (λ,μ)-fuzzy subgroup of G and λ<A(e)<μ. Then A(x)=A(e) for all x∈AA(e).Proof.
For everyx∈AA(e), A(x)⩾A(e). By Corollary 7, A(x)⩽A(e). Hence A(x)=A(e).Proposition 9 (see [4]).
LetA be a (λ,μ)-fuzzy subset of G. Then A is a (λ,μ)-fuzzy subgroup of G if and only if Aα≠∅ is a subgroup of G for all α∈(λ,μ].Proposition 10.
LetA be a (λ,μ)-fuzzy subgroup of G and x∈G. Assume(i)
if  A(x)⩾μ,  then  A(x-1)⩾μ.(ii)
If  λ<A(x)<μ,  then  A(x-1)=A(x).(iii)
If  A(x)⩽λ,  then  A(x-1)⩽λ.Proof.
SinceA is a (λ,μ)-fuzzy subgroup of G, Aα≠∅ is a subgroup of G for all α∈(λ,μ] by Proposition 9.(i)
IfA(x)⩾μ, then x∈Aμ. Since Aμ is a subgroup of G, x-1∈Aμ; that is, A(x-1)⩾μ.(ii)
Ifλ<A(x)<μ, then A(x)=A(x)∨λ⩾A(x-1)∧μ=A(x-1).IfA(x-1)∧μ=μ, then A(x-1)⩾μ; that is, A(x)⩾μ, which is contradictory to that A(x)<μ. Hence A(x)⩾A(x-1).In addition, sincex∈AA(x) and AA(x) is a subgroup of G, we have x-1∈AA(x). Therefore, A(x-1)⩾A(x). In summary, A(x-1)=A(x).(iii)
SupposeA(x-1)>λ. Let α0=min{A(x-1),μ}.
Thenλ<α0⩽μ, and A(x-1)⩾α0. Thus x-1∈Aα0. By Proposition 9, Aα0 is a subgroup of G, and thus x∈Aα0. As a result, A(x)⩾α0>λ, which is a contradiction to that A(x)⩽λ. Therefore, A(x-1)⩽λ.Proposition 11.
LetA be a (λ,μ)-fuzzy subgroup of G and x,y∈G. (i)
If  A(x)⩾μ  and  A(y)⩾μ,  then  A(xy)⩾μ.(ii)
If  λ<A(x)<μ,  A(x)<A(y),  then  A(xy)=A(x)=A(yx).(iii)
If  A(x)⩽λ,  A(y)>λ,  then  A(xy)⩽λ  and  A(yx)⩽λ.Proof.
SinceA is a (λ,μ)-fuzzy subgroup of G, Aα≠∅ is a subgroup of G for all α∈(λ,μ] by Proposition 9.(i)
FromA(x)⩾μ and A(y)⩾μ, it follows that x,y∈Aμ. Since Aμ is a subgroup of G, we have xy∈Aμ; that is, A(xy)⩾μ.(ii)
LetA(x)=α and A(y)=β, A(xy)=γ. Then λ<α<μ and β>α.Now we have the following implications successively:(1)λ<A(x)<A(y)⟹x,y∈Aα⟹xy∈Aα(byProposition9)⟹A(xy)⩾α⟹γ⩾α.Ifγ>α, let α0=min{γ,β,μ}. Then λ<α0⩽μ and xy,y∈Aα0. By Proposition 9, Aα0 is a subgroup of G, thus x=xyy-1∈Aα0. Hence A(x)⩾α0⩾γ>α, which is a contradiction. Consequently, γ=α; that is, A(xy)=A(x). Similarly, A(yx)=A(x).(iii)
SupposeA(xy)>λ. Let α0=min{A(xy),A(y),μ}.
Thenλ<α0⩽μ and xy,y∈Aα0. By Proposition 9, Aα0 is a subgroup of G, thus x=xyy-1∈Aα0. It follows that A(x)⩾α0>λ, which is a contradiction. Hence A(xy)⩽λ. Similarly, A(yx)⩽λ.Proposition 12.
LetG=(a) be a cyclic group with generator a. If A is a (λ,μ)-fuzzy subgroup of G and λ<A(a)<μ, then A(x)⩾A(a)∀x∈G.Proof.
For anyx∈G, there must be a positive integer m such that x=am. By the definition of (λ,μ)-fuzzy subgroup,
(2)A(a2)∨λ⩾A(a)∧μ=A(a),
which implies A(a2)⩾A(a). Similarly,
(3)A(a3)∨λ⩾A(a2)∧A(a)∧μ⩾A(a)∧μ=A(a),
and thus A(a3)⩾A(a). Continuing in this way, we have A(am)⩾A(a); that is, A(x)⩾A(a).Proposition 13.
IfG=(a) is a cyclic group and A(a)⩾μ, then G=Aμ.Proof.
For anyx∈G, there must be a positive integer m such that x=am. By the definition of (λ,μ)-fuzzy subgroup,
(4)A(a2)∨λ⩾A(a)∧μ=μ,
which implies A(a2)⩾μ. Similarly,
(5)A(a3)∨λ⩾A(a2)∧A(a)∧μ=μ,
which implies A(a3)⩾μ. Continuing in this way, we have A(am)⩾μ. Hence A(x)⩾μ; that is, x∈Aμ. Therefore G=Aμ.Corollary 14.
LetG be a cyclic group with generators a and b, A a (λ,μ)-fuzzy subgroup of G. If λ<A(a)<μ, then A(a)=A(b).Proof.
By Proposition12, A(b)⩾A(a). If A(b)⩾μ, then G=Aμ by Proposition 13. Particularly, A(a)⩾μ, which is a contradiction to that A(a)<μ. Hence A(b)<μ; thus λ<A(b)<μ. By Proposition 12, A(a)⩾A(b). Consequently, A(a)=A(b).Corollary 15.
LetG be a cyclic group of a prime order with generator a, A a (λ,μ)-fuzzy subgroup of G. If λ<A(a)<μ, then A(x)=A(a) for x≠e.Proof.
When the order ofG is a prime, every x≠e in G is a generator of G. By Corollary 14, A(x)=A(a) for x≠e.


## 3. Properties of(λ,μ)-Fuzzy Normal Subgroups

The notion of fuzzy normal subgroup was first proposed and investigated by Wu [6] in 1981.Definition 16 (see [6]).
LetA be a fuzzy subgroup of G. A is called a fuzzy normal subgroup of G if for all x,y∈G,
(6)A(xyx-1)⩾A(y).Proposition 17 (see [6]).
LetA be a fuzzy subgroup of G. A is a fuzzy normal subgroup if and only if for all x,y∈G,
(7)A(xy)⩾A(yx).Proposition 18 (see [6]).
LetA be a fuzzy subset of G. Then A is a fuzzy normal subgroup of G if and only if Aα≠∅ is a normal subgroup of G for all α∈(0,1].In 2005, the following notion of(λ,μ)-fuzzy normal subgroup was put forward by Yao [5].Definition 19.
LetA be a (λ,μ)-fuzzy subgroup of G. A is called a (λ,μ)-fuzzy normal subgroup of G if, for all x,y∈G,
(8)A(xyx-1)∨λ⩾A(y)∧μ.
Clearly, a (0,1)-fuzzy normal subgroup is just a fuzzy normal subgroup, and thus a (λ,μ)-fuzzy normal subgroup is a generalization of fuzzy normal subgroup.Proposition 20 (see [5]).
LetA be a (λ,μ)-fuzzy subgroup of G. A is a (λ,μ)-fuzzy normal subgroup if and only if, for all x,y∈G,
(9)A(xy)∨λ⩾A(yx)∧μ.Proposition 21 (see [5]).
LetA be a (λ,μ)-fuzzy subset of G. Then A is a (λ,μ)-fuzzy normal subgroup of G if and only if Aα≠∅ is a normal subgroup of G for all α∈(λ,μ].Proposition 22.
LetA be a (λ,μ)-fuzzy normal subgroup of G and x∈G. (i)
If  A(x)⩾μ,  then  A(yxy-1)⩾μ  for all  y∈G.(ii)
If  λ<A(x)<μ,  then  A(yxy-1)=A(x)  for all  y∈G.(iii)
If  y∈G  and  λ<A(xy)<μ,  then  A(xy)=A(yx).(iv)
If  y∈G  and  A(xy)⩾μ,  then  A(yx)⩾μ.(v)
If  y∈G  and  A(xy)⩽λ,  then  A(yx)⩽λ.Proof.
(i) IfA(x)⩾μ, then x∈Aμ. By Proposition 21, Aμ is a normal subgroup of G and thus yxy-1∈Aμ. Hence A(yxy-1)⩾μ.
(ii) LetA(x)=α. Then λ<α<μ. By Proposition 21, Aα is a normal subgroup of G. Hence yxy-1∈Aα; that is, A(yxy-1)⩾α=A(x).
SupposeA(yxy-1)>α. Set α0=min{A(yxy-1),μ}. Then λ<α0⩽μ. By Proposition 21, Aα0 is a normal subgroup of G, and thus yxy-1∈Aα0. Therefore, x=y-1(yxy-1)y∈Aα0; that is, A(x)⩾α0>α, which is a contradiction to that A(x)=α. Consequently, A(yxy-1)=A(x).
(iii) Ifλ<A(xy)<μ, then A(yx)=A(x-1(xy)x)=A(xy) by (ii); that is, A(xy)=A(yx).
(iv) IfA(xy)⩾μ, then xy∈Aμ. Since Aμ is a normal subgroup of G by Proposition 21, yx=x-1(xy)x∈Aμ; that is, A(yx)⩾μ.
(v) SupposeA(yx)>λ on the contrary.
IfA(yx)⩾μ, then, by (i), A(xy)⩾μ, which is contradictory to that A(xy)<μ. If A(yx)<μ, then, by (iii), A(xy)=A(yx)>λ, which is contradictory to that A(xy)⩽λ. Hence A(yx)⩽λ.Proposition 23.
LetA be a (λ,μ)-fuzzy subgroup of G. Then A is a (λ,μ)-fuzzy normal subgroup of G if and only if A([x,y])∨λ⩾A(x)∧μ for all x,y∈G, where [x,y]=x-1y-1xy is a commutator in G.Proof.
For anyx,y∈G,
(10)A([x,y])∨λ=A(x-1y-1xy)∨λ=A(x-1(y-1xy))∨λ∨λ⩾(A(x-1)∧A(y-1xy)∧μ)∨λ=(A(x-1)∨λ)∧(A(y-1xy)∨λ)∧μ.
Since A is a (λ,μ)-fuzzy normal subgroup of G, A(x-1)∨λ⩾A(x)∧μ and A(y-1xy)∨λ⩾A(x)∧μ. Therefore,
(11)A([x,y])∨λ⩾A(x)∧μ.
Conversely, if A([x,y])∨λ⩾A(x)∧μ, then
(12)A(y-1xy)∨λ=A(xx-1y-1xy)∨λ=A(x[x,y])∨λ∨λ⩾(A(x)∧A([x,y])∧μ)∨λ=(A(x)∨λ)∧(A([x,y])∨λ)∧μ⩾(A(x)∨λ)∧(A(x)∧μ)∧μ⩾A(x)∧A(x)∧μ=A(x)∧μ.
Hence A is a (λ,μ)-fuzzy normal subgroup of G.Proposition 24.
IfG is an abelian group and A is a (λ,μ)-fuzzy subgroup of G, then A is a (λ,μ)-fuzzy normal subgroup of G.Proof.
SinceG is an abelian group, we have [xy]=e; hence A([xy])∨λ=A(e)∨λ⩾A(x)∧μ for all x,y∈G by Proposition 5. By Proposition 23, A is a (λ,μ)-fuzzy normal subgroup of G.Since a cyclic group is an abelian group, the following result is immediate by Proposition24.Corollary 25.
IfG is a cyclic group and A is a (λ,μ)-fuzzy subgroup of G, then A is a (λ,μ)-fuzzy normal subgroup of G.


## 4. Properties of Left Cosets and Right Cosets ofA

Definition 26 (see [5]).
LetA be a (λ,μ)-fuzzy subgroup of G and a∈G. Define fuzzy subsets a∘A and A∘a of G respectively by
(13)(a∘A)(x)=(A(a-1x)∨λ)∧μ,∀x∈G,(A∘a)(x)=(A(xa-1)∨λ)∧μ,∀x∈G.
a
∘
A and A∘a will be called a left coset and a right coset of A, respectively.Clearly, we havee∘A=A∘e, λ⩽(A∘a)(x)⩽μandλ⩽(a∘A)(x)⩽μ which are valid for all x∈G.The following conclusions can be found in [5].Proposition 27 (see [5]).
LetA and B be (λ,μ)-fuzzy subgroups of G and a,b∈G. Then (i)
a∘(b∘A)=(ab)∘A.(ii)
(A∘a)∘b=A∘(ab).(iii)
a∘A=b∘B⇔e∘A=(a-1b)∘B⇔(b-1a)∘A=e∘B.(iv)
A∘a=B∘b⇔A∘e=B∘(ba-1)⇔A∘(ab-1)=B∘e.In particular, we have the following corollary whenA=B.Corollary 28.
LetA be a (λ,μ)-fuzzy subgroup of G and a,b∈G. Then (i)
a∘A=b∘A⇔e∘A=(a-1b)∘A⇔(b-1a)∘A=e∘A;(ii)
A∘a=A∘b⇔A∘e=A∘(ba-1)⇔A∘(ab-1)=A∘e.Firstly, we present some basic properties ofe∘A.Proposition 29.
LetA be a (λ,μ)-fuzzy subgroup of G and x∈G.(i)
If  λ<A(x)<μ,  then  (e∘A)(x)=A(x).(ii)
If  A(x)⩾μ,  then  (e∘A)(x)=μ.(iii)
If  A(x)⩽λ,  then  (e∘A)(x)=λ.Proof.
(i) Ifλ<A(x)<μ, then (e∘A)(x)=(A(x)∨λ)∧μ=A(x). (ii) and (iii) can be similarly proved.Proposition 30.
LetA be a (λ,μ)-fuzzy subgroup of G. Then e∘A is a (λ,μ)-fuzzy subgroup of G and a fuzzy subgroup of G in the sense of Rosenfeld as well.Proof.
Firstly, we prove thatAα=(e∘A)α for all α∈(λ,μ].
Ifx∈Aα, then A(x)⩾α>λ. Hence,
(14)(e∘A)(x)=(A(x)∨λ)∧μ=A(x)∧μ⩾α,
which implies x∈(e∘A)α. So Aα⊆(e∘A)α.
Conversely, ifx∈(e∘A)α, then (e∘A)(x)=(A(x)∨λ)∧μ⩾α, whence A(x)⩾α; that is, x∈Aα. Thus, Aα⊇(e∘A)α. In summary, Aα=(e∘A)α.
SinceA is a (λ,μ)-fuzzy subgroup of G, Aα≠∅ is the subgroup of G by Proposition 9. It follows from Aα=(e∘A)α that (e∘A)α is a subgroup of G. Hence e∘A is a (λ,μ)-fuzzy subgroup of G by Proposition 9.
Sinceλ⩽(e∘A)(x)⩽μ,∀x∈G, (e∘A)α=G for α∈(0,λ], and clearly it is a subgroup of G. Considering that (e∘A)α=∅ for α∈(μ,1], (e∘A)α≠∅ is subgroup of G for all α∈(0,1]. As a result, e∘A is a fuzzy subgroup of G in the sense of Rosenfeld by Proposition 4.Proposition 31.
LetA be a (λ,μ)-fuzzy normal subgroup of G. Then e∘A is a (λ,μ)-fuzzy normal subgroup of G and a fuzzy normal subgroup of G as well.Proof.
By Proposition30, e∘A is a (λ,μ)-fuzzy subgroup of G and a fuzzy subgroup of G. For any x,y∈G,
(15)(e∘A)(xyx-1)∨λ=(e∘A)(xyx-1)=(A(xyx-1)∨λ)∧μ=((A(xyx-1)∨λ)∨λ)∧μ⩾((A(y)∧μ)∨λ)∧μ=(A(y)∨λ)∧μ=(e∘A)(y)=(e∘A)(y)∧μ.
Therefore, e∘A is a (λ,μ)-fuzzy normal subgroup of G by Definition 19 and a fuzzy normal subgroup of G by Definition 16.Furthermore, we have the following results.Proposition 32.
IfA is a (λ,μ)-fuzzy subgroup of G and A(e)⩾μ, then a∈Aμ if and only if a∘A=e∘A.Proof.
Firstly, we assumea∈Aμ. Since Aμ is a subgroup of G by Proposition 9, we have a-1∈Aμ; that is, A(a-1)⩾μ by Proposition 10. Let x∈G. Consider the following three cases.
Case  1. If A(x)⩾μ, then x∈Aμ, which implies a-1x∈Aμ; that is, A(a-1x)⩾μ by Proposition 11(i). Therefore,
(16)(a∘A)(x)=(A(a-1x)∨λ)∧μ=μ=(A(x)∨λ)∧μ=(e∘A)(x).
Case  2. If λ<A(x)<μ, then A(a-1x)=A(x) by Proposition 11(ii). Hence
(17)(a∘A)(x)=(A(a-1x)∨λ)∧μ=(A(x)∨λ)∧μ=(e∘A)(x).
Case  3. If A(x)⩽λ, then A(a-1x)⩽λ by Proposition 11(iii). Hence
(18)(a∘A)(x)=(A(a-1x)∨λ)∧μ=λ=(A(x)∨λ)∧μ=(e∘A)(x).
In summary, a∘A=e∘A.
Conversely, assumea∘A=e∘A. Then we have the following implications successively:
(19)(a∘A)(a)=(e∘A)(a)⟹(A(a-1a)∨λ)∧μ=(A(a)∨λ)∧μ⟹(A(e)∨λ)∧μ=(A(a)∨λ)∧μ⟹μ=(A(a)∨λ)∧μ(sinceA(e)⩾μ)⟹A(a)⩾μ.⟹a∈Aμ.Similarly, ifA is a (λ,μ)-fuzzy subgroup of G and A(e)⩾μ, then a∈Aμ if and only if A∘a=A∘e.Corollary 33.
IfA is a (λ,μ)-fuzzy subgroup of G and a∈Aμ, then
(20)a∘A=A∘a=e∘A.Proof.
Sincea∈Aμ, we have A(e)⩾μ by Corollary 6, and hence a∘A=e∘A and A∘a=A∘e. It follows from e∘A=A∘e that a∘A=A∘a=e∘A.Corollary 34.
LetA be a (λ,μ)-fuzzy subgroup of G and a,b∈G. Then aAμ=bAμ if and only if a∘A=b∘A provided Aμ≠∅.Proof.
The desired result follows from the following equivalences:(21)aAμ=bAμ⟺b-1a∈Aμ⟺(b-1a)∘A=e∘A(byProposition32)⟺a∘A=b∘A(byCorollary28).Similarly, under the conditions stated in Corollary34, we have the equivalence Aμa=Aμb if and only if A∘a=A∘b.Proposition 35.
IfA is a (λ,μ)-fuzzy subgroup of G and λ<A(e)<μ, then A(a)=A(e) if and only if a∘A=e∘A.Proof.
Firstly, we assumeA(a)=A(e); that is, a∈AA(e). By Proposition 9, AA(e) is a subgroup of G. Hence we have a-1∈AA(e); that is, A(a-1)⩾A(e), so A(a-1)=A(e) by Corollary 6. Let x∈G. We have A(x)⩽A(e) by Corollary 6. Consider the following three cases.
Case  1  (A(x)⩽λ).  In this case, A(a-1x)⩽λ by Proposition 11(iii). So
(22)(a∘A)(x)=(A(a-1x)∨λ)∧μ=λ=(A(x)∨λ)∧μ=(e∘A)(x).Case  2  (λ<A(x)<A(e)=A(a-1)).  In this case, A(a-1x)=A(x) by Proposition 11(ii). So
(23)(a∘A)(x)=(A(a-1x)∨λ)∧μ=(A(x)∨λ)∧μ=(e∘A)(x).Case  3  (A(x)=A(e)).  Since AA(e) is a subgroup of G by Proposition 9, we have x,a-1∈AA(e). Hence a-1x∈AA(e); that is, A(a-1x)⩾A(e). Thus A(a-1x)=A(e) by Corollary 6. As a result, A(a-1x)=A(x)=A(e). Therefore,
(24)(a∘A)(x)=(A(a-1x)∨λ)∧μ=(A(x)∨λ)∧μ=(e∘A)(x).
In summary, a∘A=e∘A.
Conversely, assumea∘A=e∘A. Then we have the following implications successively:
(25)(a∘A)(a)=(e∘A)(a)⟹(A(a-1a)∨λ)∧μ=(A(a)∨λ)∧μ⟹A(e)=A(a)∨λ(sinceλ<A(e)<μ)⟹A(e)=A(a).Similarly, ifA is a (λ,μ)-fuzzy subgroup of G and λ<A(e)<μ, then A(a)=A(e) if and only if A∘a=A∘e.Corollary 36.
LetA be a (λ,μ)-fuzzy subgroup of G and a,b∈G. Then aAA(e)=bAA(e) if and only if a∘A=b∘A provided λ<A(e)<μ.Proof.
The desired result follows from the following equivalencies:(26)aAA(e)=bAA(e)⟺b-1a∈AA(e)⟺(b-1a)∘A=e∘A(byProposition35)⟺a∘A=b∘A(byCorollary28).Similarly, under the conditions stated in Corollary36, we have the equivalence AA(e)a=AA(e)b if and only if A∘a=A∘b.Proposition 37.
LetA be a (λ,μ)-fuzzy subgroup of G. Then a∘A is a (λ,μ)-fuzzy subgroup of G if and only if a∘A=e∘A.Proof.
Suppose thata∘A is a (λ,μ)-fuzzy subgroup of G.
IfA(e)⩾μ, then (a∘A)(a)=(A(e)∨λ)∧μ=μ, which implies a∈(a∘A)μ, and thus (a∘A)μ≠∅. Since (a∘A)μ is a subgroup of G by Proposition 9, we have e∈(a∘A)μ. Hence (a∘A)(e)=(A(a-1)∨λ)∧μ=μ, which implies A(a-1)⩾μ; that is, a-1∈Aμ. Hence we have a∈Aμ by Proposition 10(i). Therefore, a∘A=e∘A by Proposition 32.
Ifλ<A(e)<μ, then A(e)⩾A(a-1) by Corollary 6 and (a∘A)(a)=(A(e)∨λ)∧μ=A(e), which implies a∈(a∘A)A(e). Since a∘A is a (λ,μ)-fuzzy subgroup of G, (a∘A)A(e) is a subgroup of G by Proposition 9, which implies e∈(a∘A)A(e). Hence
(27)A(e)≤(a∘A)(e)=(A(a-1)∨λ)∧μ=A(a-1)∨λ,
and thus A(a-1)⩾A(e); that is, a-1∈AA(e), which implies a∈AA(e) since AA(e) is a subgroup of G. Therefore, A(a)⩾A(e), and hence A(a)=A(e) by Corollary 6. It follows a∘A=e∘A from Proposition 35.
IfA(e)⩽λ, then A(x)⩽λ for all x∈G by Corollary 6. Consider
(28)(a∘A)(x)=(A(a-1x)∨λ)∧μ=λ=(A(x)∨λ)∧μ=(e∘A)(x).
Hence a∘A=e∘A.
Conversely, ifa∘A=e∘A, then a∘A is a (λ,μ)-fuzzy subgroup of G by Proposition 30.Proposition 38.
LetA be a (λ,μ)-fuzzy subgroup of G, a,b∈G, α∈(λ,μ] and Aα≠∅. If a∘A=b∘A, then aAα=bAα.Proof.
SinceAα≠∅, Aα is a subgroup of G by Proposition 9, and thus A(e)⩾α>λ. Hence, from a∘A=b∘A, we know that (a∘A)(b)=(b∘A)(b); that is,
(29)(A(a-1b)∨λ)∧μ=(A(e)∨λ)∧μ=A(e)∧μ.
If A(e)⩾μ, then A(a-1b)⩾μ; that is, a-1b∈Aμ⊆Aα. Hence aAα=bAα.
IfA(e)<μ, then A(a-1b)=A(e); that is, a-1b∈AA(e). By AA(e)⊆Aα, thus a-1b∈Aα. Hence aAα=bAα.Similarly, we have the following conclusion.LetA be a (λ,μ)-fuzzy subgroup of G, a,b∈G, α∈(λ,μ] and Aα≠∅. If A∘a=A∘b, then Aαa=Aαb.Proposition 39.
LetA be a (λ,μ)-fuzzy subgroup of G and a,b∈G. If a∘A=b∘A and λ<A(a)<μ, then A(a)=A(b).Proof.
Sincea∘A=b∘A, (a-1b)∘A=e∘A by Corollary 28.
Case  1. If A(e)⩾μ, then by Proposition 32, a-1b∈Aμ; that is, A(a-1b)⩾μ. By Proposition 11 and λ<A(a)<μ, A(a)=A(a(a-1b))=A(b).
Case  2. If A(e)<μ and λ<A(a)<μ, then A(a)⩽A(e)<μ and then A(a-1b)=A(e) by Proposition 35.
IfA(a)<A(e)=A(a-1b), then by Proposition 11, A(a)=A(a(a-1b))=A(b).
IfA(a)=A(e)=A(a-1b), then a,a-1b∈AA(e). Since A is a (λ,μ)-fuzzy subgroup of G, AA(e) is a subgroup of G by Proposition 9. Hence b=a(a-1b)∈AA(e); that is, A(b)⩾A(e); hence A(b)=A(e)=A(a) by Corollary 7.It is noteworthy that the converse of Proposition39 is not true as shown by the following example.Example 40.
LetG be a cyclic group of order 3 with generator a. Define A by: A(a)=A(a2)=0.5, A(e)=0.7. Then A is a (0.3,0.6)-fuzzy subgroup of G. In this example, however, (a∘A)(a)=0.6, (a2∘A)(a)=0.5. Hence a∘A≠a2∘A.


## 5. Concluding Remarks

In this paper, we present a further investigation into properties of(λ,μ)-fuzzy subgroups, (λ,μ)-fuzzy normal subgroups, and (λ,μ)-fuzzy left and right cosets. It should be noted that we concentrate on the case A(e)>λ in studying fuzzy left and right cosets. The reason is that (a∘A)(x)=λ is valid for all x∈G when A(e)⩽λ, which is of little mathematical significance. Certainly, some other topics of (λ,μ)-fuzzy subgroups such as the characterization of (λ,μ)-fuzzy subgroups and operations of (λ,μ)-fuzzy subgroups are still open for the research in the future.



---

*Source: 485768-2013-11-27.xml*