# On the Algebraic Characteristics of Fuzzy Sub e-Groups

Authors: Supriya Bhunia; Ganesh Ghorai; Marwan Amin Kutbi; Muhammad Gulzar; Md Ashraful Alam
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Fuzzy set is a modern tool for depicting uncertainty. This paper introduces the concept of fuzzy sub e-group as an extension of fuzzy subgroup. The concepts of identity and inverse are generalized in fuzzy sub e-groups. Every fuzzy subgroup is proven to be a fuzzy sub e-group, but the converse is not true. Various properties of fuzzy sub e-groups are established. Moreover, the concepts of proper fuzzy sub e-group and super fuzzy sub e-group are discussed. Further, the concepts of fuzzy e-coset and normal fuzzy sub e-group are presented. Finally, we describe the effect of e-group homomorphism on normal fuzzy sub e-groups.


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## Body

## 1. Introduction

Many decades ago, researchers developed an algebraic structure and named it as a group. Various properties of groups are proposed later on. In group theory, every group contains a unique identity element and every element has a unique inverse. In 2018, Saeid et al. [1] generalized the notion of groups to a new algebraic structure as e-groups. They generalized the notion of the identity of a group. Instead of choosing a single element as an identity element, Saeid et al. [1] considered a subset of the main set as an identity set. So, in an e-group, the identity element needs not to be unique. They proved that every group is an e-group, but the converse is not true. They defined homomorphism on e-groups in a different manner. E-group is an important tool for classifying isotopes. It is also a physical background in the unified Gauge theory.Uncertainty is a massive component in the life of a person. In 1965, in his pioneer paper, Zadeh [2] first defined fuzzy set to handle uncertainty in real-life problems. In 1971, utilizing the concept of fuzzy set, Rosenfeld [3] first defined fuzzy subgroup. In 1979, using the t-norm concept of the fuzzy subgroup was restructured by Anthony and Sherwood [4, 5]. In 1981, the idea of fuzzy level subgroup was introduced by Das [6]. In 1988, Choudhury et al. [7] proved various properties of fuzzy homomorphism. In 1990, Dixit et al. [8] discussed the union of fuzzy subgroups and fuzzy level subgroups. The concept of antifuzzy subgroups was proposed by Biswas [9]. In 1992, Ajmal and Prajapati [10] developed fuzzy cosets and fuzzy normal subgroups. Chakraborty and Khare [11] studied various properties of fuzzy homomorphism. Ajmal [12] also studied the homomorphism of fuzzy subgroups. Later, many researchers studied various properties of fuzzy subgroups [13–16]. In 2015, Tarnauceanu [17] classified fuzzy normal subgroups of finite groups. In 2016, Onasanya [18] reviewed some antifuzzy properties of fuzzy subgroups. In 2018, Shuaib and Shaheryar [19] introduced omicron fuzzy subgroups. In 2018, Addis [20] developed fuzzy homomorphism theorems on groups. In 2019, Bhunia and Ghorai [21] studied α,β-Pythagorean fuzzy subgroups. In 2021, Bhunia et al. [22, 23] developed Pythagorean fuzzy subgroups. Abuhijleh et al. [24] worked on complex fuzzy subgroups in 2021. Alolaiyan et al. [25] studied algebraic structure of α,β-complex fuzzy subgroups. Alolaiyan et al. [26] developed bipolar fuzzy subrings in 2021. In 2021, Talafha et al. [27] studied fuzzy fundamental groups and fuzzy folding of fuzzy Minkowski space.In a fuzzy subgroup, the identity element of the group has the highest membership value. Also, in a fuzzy subgroup, the membership value of an element and its inverse are equal. But in an e-group, there is no unique identity element and no direct concept of the inverse of elements. Till now, no fuzzification has been done for e-groups. So, it is a challenge for us to fuzzify e-groups. In this study, we construct the concept of fuzzy sub e-groups. Here, we show that every fuzzy subgroup is a fuzzy sub e-group, but the converse is not true. So, the idea of fuzzy sub e-group is a much more generalized concept. In the study of isotopes, we notice that isotopes decay through neutron emission. So, the membership degree of these unstable neutrons must lie in0,1. Therefore, fuzzy sub e-group will be more efficient in studying these unstable isotopes rather than a crisp e-group.This paper is arranged in the following order. In Section2, we recall some important concepts. In Section 3, utilizing the concept of e-groups, we generalize fuzzy subgroups. We develop the concept of fuzzy sub e-group and show that any fuzzy subgroup is also a fuzzy sub e-group. We also prove many algebraic properties of fuzzy sub e-groups. Further, we define fuzzy e-coset and normal fuzzy sub e-group in Section 4. Moreover, in Section 5, we show that after e-group homomorphism, a fuzzy sub e-group remains a fuzzy sub e-group. Finally, the conclusion is given in Section 6.


## 2. Preliminaries

Here, we will go over some basic definitions and concepts, which will be useful in the following sections.Definition 1 (see [2]).
A fuzzy set (FS)D,κ on a crisp set D is an object having the form D,κ=d,κdd∈D, where κ⟶0,1 is the membership function.Definition 2 (see [3]).
LetD,κ be a FS on a group D. Then, D,κ is referred to be a fuzzy subgroup (FSG) of D if the following conditions hold:
(i)
κd1d2≥κd1∧κd2∀d1,d2∈D(ii)
κd−1≥κd∀d∈DDefinition 3 (see [2]).
LetD,κ be a FS on D. Then, for any a∈0,1, the set κa=d∣d∈D,κd≥a is called a-cut of D,κ.Clearly,κa is a subset of D.Proposition 4 (see [28]).
Leth be a mapping from D1 into D2. Let D1,κ1 and D2,κ2 be the two FSs on D1 and D2, respectively. Then, D2,hκ1 and D1,h−1κ2 are FSs on D2 and D1, respectively, where for all d2∈D2(1)hκ1d2=∨κ1d1d1∈D1,hd1=d2,whenh−1d2≠∅,0,elsewhere,and for all d1∈D1, h−1κ2d1=κ2hd1.Definition 5 (see [1]).
LetD be a nonempty crisp set and L⊆D. Then, D,∘,L is an e-group, where ∘ is the binary operation on D, which meets the following criteria:
(i)
d1∘d2∘d3=d1∘d2∘d3∀d1,d2,d3∈D(ii)
For everyd∈D, ∃ an element l∈L such that d∘l=l∘d=d(iii)
For everyd1∈D, ∃ an element d2∈D such that d1∘d2 and d2∘d1∈LDefinition 6 (see [1]).
LetD1,∘1,L1 and D2,∘2,L2 be the two e-groups. If a mapping h:D1⟶D2 meets the following criteria, it is referred to as a homomorphism:
(i)
hL1⊆L2(ii)
hd1∘1d2=hd1∘2hd2∀d1,d2∈D1


## 3. Fuzzy Sub e-Group and Its Properties

In this section, fuzzy sub e-group is briefly described as a generalization of fuzzy subgroup. The notions of identity and inverse are generalized in fuzzy sub e-group. We investigate its properties. We define super fuzzy sub e-group. We check whether union and intersections of fuzzy sub e-group are fuzzy sub e-groups.Definition 7.
A FSD,κ is referred to be a fuzzy sub e-group of an e-group D,∘,L if the following conditions hold:
(i)
κd1∘d2≥κd1∧κd2∀d1,d2∈D(ii)
κl≥κd∀l∈L and d∈D/Lwhere κ:D⟶0,1 is the membership function.Example 8.
LetD=d1,d2,d3 and L=d1,d2. Define a binary operation ∘ on D as below.
(2)
Then,D,∘,L is an e-group.
Here, we assign membership degrees to the elements ofD by κd1=κd3=0.8 and κd2=0.9.
Now,κd1∘d3=κd2=0.9>0.8=κd1∧κd3.
Similarly, we can check for other elements ofD.
Therefore,κdi∘dj≥κdi∧κdj for all di, dj∈D.
Also,κd1=0.8=κd3 and κd2=0.9>0.8=κd3.
Thus,D,κ forms a fuzzy sub e-group of the e-group D,∘,L.Theorem 9.
LetD,∘,L stand for an e-group and D,κ be a FS on D. Then, D,κ is referred to be a fuzzy sub e-group of D,∘,L if for all d1 and d2∈D, κd1∘d2′≥κd1∧κd2 for some d2′∈D such that d2∘d2′, d2′∘d2∈L.Proof.
LetD,κ stand for a fuzzy sub e-group of D,∘,L.
Then, for alld1 and d2 in D, κd1∘d2≥κd1∧κd2.
Letd1, d2∈D. Then, ∃ some d2′∈D such that d2∘d2′ and d2′∘d2∈L.
Therefore,κd1∘d2′≥κd1∧κd2′≥κd1∧κd2.
Conversely, assume that∀d1,d2∈D, κd1∘d2′≥κd1∧κd2 for some d2′∈D such that d2∘d2′ and d2′∘d2∈L. Let d1∈D, d2∈D/L.
Here,d2∘d2′∈E, then κd2∘d2′≥κd2∧κd2=κd2.
Therefore,κl≥κd, where l∈L and d∈D/L.
Letd2=d2′′ for all d2, d2′∈D such that d2∘d2′ and d2′∘d2∈L.
Therefore,d1∘d2=κd1∘d2′′≥κd1∧κd2′≥κd1∧κd2∀d1,d2∈D.
Hence, the e-groupD,∘,L has a fuzzy sub e-group D,κ.Remark 10.
The above theorem gives the necessary and sufficient condition for a FS of an e-group to be a fuzzy sub e-group.Now, we will demonstrate that any FSG within a groupD is also a fuzzy sub e-group of the e-group D,∘,l, where l is the group’s identity element. But the converse needs not to be true.Theorem 11.
Any fuzzy subgroup of a group is a fuzzy sub e-group.Proof.
LetD,κ stand for a FSG of a group D.
Then,κd1∘d2≥κd1∧κd2 and κd1−1=κd1 for all d1 and d2∈D.
So, the first condition of fuzzy sub e-group is satisfied.
AsD is a group, then there is a unique identity element l in D.
SinceD,κ is a FSG of D, κl≥κd∀d∈D.
Now, we takeL=l; then, L⊆D.
Therefore,κl≥κd, where l∈L and d∈D/L.
Hence, the FSGD,κ of D is a fuzzy sub e-group of the e-group D,·,L.Example 12.
LetD=d1,d2,d3,d4 and L=d1,d2.
Define∘ on D as binary operation by the following:
(3)
Then,D,∘,L is an e-group.
Now, we assign a membership value to each of the elements ofD by the following:
(4)κd1=0.8,κd2=0.9,κd3=0.6,κd4=0.7.
Now, we can verify thatD,κ is a fuzzy sub e-group of the e-group D,∘,L.
ButL>1. So, D,∘ is not a group. Hence, the FS D,κ is not a FSG.Remark 13.
A fuzzy sub e-group of an e-group is not necessarily a FSG.Definition 14.
A fuzzy sub e-group of an e-group which is not a FSG is said to be a proper fuzzy sub e-group.
The fuzzy sub e-groupD,κ in Example 12. is a proper fuzzy sub e-group.
Now, we will check about union and intersection of fuzzy sub e-groups.Theorem 15.
Intersection of fuzzy sub e-groups of an e-group is also a fuzzy sub e-group of that e-group.Proof.
LetD,κ1 and D,κ2 be the two fuzzy sub e-groups of an e-group D,∘,L.
Then,∀d1,d2∈D, κ1d1∘d2≥κ1d1∧κ1d2 and κ1l≥κ1d, where l∈L and d∈D/L.
Also,∀d1,d2∈D, κ2d1∘d2≥κ2d1∧κ2d2 and κ2l≥κ2d, where l∈L and d∈D/L.
LetD,κ be the intersection of D,κ1 and D,κ2, where κ=κ1∩κ2 is given by κd=κ1d∧κ2d∀d∈D.
Now for alld1,d2∈D,
(5)κd1∘d2=κ1d1∘d2∧κ2d1∘d2≥κ1d1∧κ1d2∧κ2d1∧κ2d2=κ1d1∧κ2d1∧κ1d2∧κ2d2=κd1∧κd2.
Therefore,κd1∘d2≥κd1∧κd2 for all d1,d2∈D.
Again forl∈L and d∈D/L, we have the following:
(6)κl=κ1l∧κ2l≥κ1d∧κ2d=κd.
Therefore,D,κ is a fuzzy sub e-group of the e-group D,∘,L.
Hence, the intersection of two fuzzy sub e-groups of an e-group is also a fuzzy sub e-group of that e-group.Corollary 16.
Intersection of any fuzzy sub e-groups of an e-groupD,∘,L is also a fuzzy sub e-group of that e-group D,∘,L.Remark 17.
Union of two fuzzy sub e-groups of an e-group may not be a fuzzy sub e-group of that e-group.Example 18.
Let us take the e-groupD,∘,L, where D=ℤ, L=2ℤ, and ∘ is the addition of integers.
LetD,κ1 and D,κ2 be the two fuzzy sub e-groups of the e-group D,∘,L, where κ1 and κ2 are presented by the following:
(7)κ1d=0.6,whend∈2ℤ,0.3,whend∈5ℤ/2ℤ,0,elsewhere,κ2d=0.8,whend∈2ℤ,0.2,whend∈3ℤ/2ℤ,0,elsewhere.
LetD,κ be the union of D,κ1 and D,κ2, where κ=κ1∪κ2 is given by κd=κ1d∨κ2d∀d∈D.
Therefore,(8)κd=0.8,whend∈2ℤ,0.3,whend∈5ℤ/2ℤ,0.2,whend∈3ℤ/2ℤ∩5ℤ,0,elsewhere.
Now,κ5+−4=κ1=0, but κ5∧κ−4=min0.3,0.8=0.3.
So,κ5+−4≱κ5∧κ−4.
Hence,D,κ is not a fuzzy sub e-group of the e-group D,∘,L.Definition 19.
LetD,κ1 and D,κ2 be the two fuzzy sub e-groups of an e-group D,∘,L such that κ2l≥κ1l for all l∈L and κ2d≤κ1d for all d∈D/L, then D,κ2 is referred to be a super fuzzy sub e-group of D,κ1.Example 20.
In Example12., we take another FS D,κ1 on D,∘,L, where
(9)κ1d1=0.85,κ1d2=0.93,κ1d3=0.57,κ1d4=0.68.
Then, we can simply verify thatD,∘,L has a fuzzy sub e-group D,κ1.
Now, we can see that for alll∈L, κ1l≥κl and for all d∈D/L, κ1d≤κd.
Hence,D,κ1 is a super fuzzy sub e-group of D,κ.Theorem 21.
LetD,κ stand for a fuzzy sub e-group of an e-group D,∘,L. Then, the set K=d∣d∈D,κd=p forms a sub e-group K,∘,L of the e-group D,∘,L, where p=∧κll∈L.Proof.
GivenK=d∣d∈D,κd=p, where p=∧κll∈L.
To show that the e-groupD,∘,L has a sub e-group K,∘,L, we have to show that K,∘,L itself forms an e-group.
SinceD,∘,L is an e-group, the associative law holds.
Clearly,K is a subset of D. Then, associative law also holds in K. Instead of showing the other two conditions of e-group, we will show that for all k1 and k2∈K, ∃ a k2′∈K such that k1∘k2′ and k2′∘k1∈L.
Letk1, k2, and k2′∈K. Then, κk1=κk2=κk2′=p.
SinceD,κ is a fuzzy sub e-group of D,∘,L, by Theorem 9, we have the following:
(10)κk1∘k2′≥κk1∧κk2=p.
Similarly, we can show thatκk2′∘k1≥p.
Sincep=∧κll∈L, k1∘k2′ and k2′∘k1∈L.
Hence,K,∘,L forms a sub e-group of D,∘,L.


## 4. Normal Fuzzy Sub e-Group and Level Fuzzy Sub e-Group

This section will describe fuzzy e-cosets and normal fuzzy sub e-groups. We will also introduce the concept of level fuzzy sub e-groups.Definition 22.
LetD,κ stand for a fuzzy sub e-group of an e-group D,∘,L. Then, ∀s,d∈D, the left fuzzy e-coset sκ=κls∘κ is defined by sκd=κs′∘d and the right fuzzy e-coset κs=κ∘κls is defined by κsd=κd∘s′, where l is any element of L and s′∈D such that s∘s′ and s′∘s∈L.If a left fuzzy e-coset is also a right fuzzy e-coset, then we will simply call it is a fuzzy e-coset.Definition 23.
Letd,κ stand for a fuzzy sub e-group of an e-group D,∘,L. Then, D,κ forms a normal fuzzy sub e-group of the e-group D,∘,L if every left fuzzy e-coset of D,κ is a right fuzzy e-coset of D,κ in D,∘,L.
Equivalently,sκ=κs for all s∈D.Example 24.
Let us take the e-groupℤ,+,2ℤ. Now, ℤ,κ forms a fuzzy sub e-group on ℤ, where κ is presented by the following:
(11)κz=0.9,whenz∈2ℤ,0.6,elsewhere.
Let us takes=3∈ℤ.
Then,∀d∈ℤ, the left fuzzy e-coset 3κ is presented by 3κd=κ3′+d=κ−3+d and the right fuzzy e-coset κ3 is presented by κ3d=κd+3′=κd−3.
Since addition is commutative onℤ, then 3κ=κ3. Similarly, we can check for other elements of ℤ. Hence, ℤ,κ forms a normal fuzzy sub e-group of ℤ,+,2ℤ.In the next theorem, we represent the necessary and sufficient condition for a fuzzy sub e-group to be a normal fuzzy sub e-group.Theorem 25.
LetD,κ stand for a fuzzy sub e-group of an e-group D,∘,L. Then, D,κ forms a normal fuzzy sub e-group of the e-group D,∘,L iff κd1∘d2=κd2∘d1∀d1,d2∈D.Proof.
LetD,κ stand for a normal fuzzy sub e-group of the e-group D,∘,L.
Then, every left fuzzy e-coset ofD,κ is also a right fuzzy e-coset of D,κ in D,∘,L.
Therefore,tκ=κt for all t∈D. That is tκd2=κtd2∀d2,t∈D.
This suggests thatκt′∘d2=κd2∘t′∀t,t′,d2∈D such that t∘t′,t′∘t∈L.
Now, we putt′=d1∈D. Therefore, κd1∘d2=κd2∘d1∀d1,d2∈D.
Conversely, letκd1∘d2=κd2∘d1∀d1,d2∈D.
Lets∈D. Since D,∘,L is an e-group, then ∃ a s′∈D such that s∘s′,s′∘s∈L.
Supposes′=d1. Then, κs′∘d2=κd2∘s′∀d2,s′∈D.
Therefore,sκd2=κsd2∀s,d2∈D. Thus, sκ=κs∀s∈D.
Hence,D,κ forms a normal fuzzy sub e-group of D,∘,L.Theorem 26.
LetD,κ stand for a fuzzy sub e-group of an e-group D,∘,L. Then, the a-cut κa of D,κ forms a sub e-group κa,∘,L of the e-group D,∘,L, where a≤∧κl∣l∈L.Proof.
We haveκa=d∣d∈D,κd≥a, where a∈0,1 and a≤∧κl∣l∈L.
Clearly,κa is nonempty as L⊆κa.
To show thatκa,∘,L is an e-subgroup of D,∘,L, we need to prove that for d1,d2,d2′∈κa, d1∘d2′∈κa, where d2∘d2′,d2′∘d2∈L.
Letd1,d2,d2′∈κa and d2∘d2′,d2′∘d2∈L. Then, κd1≥a and κd2≥a.
SinceD,κ forms a fuzzy sub e-group of the e-group D,∘,L,
(12)κd1∘d2′≥κd1∧κd2′≥κd1∧κd2≥a∧a=a.
Therefore,d1∘d2′∈κa, where d2∘d2′,d2′∘d2∈L. Hence, κa,∘,L forms a sub e-group of the e-group D,∘,L.Definition 27.
The sub e-groupκa,∘,L of the e-group D,∘,L is referred to as a level fuzzy sub e-group of D,κ.Example 28.
We consider the fuzzy sub e-groupD,κ of the e-group D,∘,L in Example 12..
Choosea=0.65. Then, κa=d1,d2,d4 and L=d1,d2.
Clearly,κa⊆D.
We can easily check thatκa,∘,L is a sub e-group of the e-group D,∘,L.
Therefore,κa,∘,L is a level fuzzy sub e-group of D,κ.


## 5. Homomorphism of Fuzzy Sub e-Groups

We shall demonstrate some important theorems on fuzzy sub e-group homomorphism in this section.Theorem 29.
LetD1,∘1,L1 and D2,∘2,L2 be the two e-groups. Let h be a bijective homomorphism from D1,∘1,L1 to D2,∘2,L2 and D2,κ be a fuzzy sub e-group of D2,∘2,L2. Then, D1,h−1κ forms a fuzzy sub e-group of D1,∘1,L1.Proof.
Letd1 and l1 be the two elements of D1. Now,
(13)h−1κd1∘1l1=κhd1∘1l1=κhd1∘2hl1≥κhd1∧κhl1=h−1κd1∧h−1κl1.
Therefore,h−1κd1∘1l1≥h−1κd1∧h−1κl1 for all d1 and l1∈D1.
Letl∈L1 and d∈D1/L1.
Sinceh is a homomorphism, hl∈L2 as hL1⊆L2. Now,
(14)h−1κl=κhl≥κhd=h−1κd..☐Therefore,h−1κl≥h−1κd for all l∈L1 and d∈D1/L1.Hence,D1,h−1κ forms a fuzzy sub e-group of D1,∘1,L1.Theorem 30.
LetD1,∘1,L1 and D2,∘2,L2 be the two e-groups. Let h be a homomorphism from D1,∘1,L1 to D2,∘2,L2 and D1,κ be a fuzzy sub e-group of D1,∘1,L1. Then, D2,hκ forms a fuzzy sub e-group of D2,∘2,L2.Proof.
Letd2 and l2 be the two elements of D2.
If eitherd2∉hD1 or l2∉hD1 then,
(15)hκd2∧hκl2=0≤hκd2∘2l2.
Supposed2=hd1 and l2=hl1 for some d1, l1∈D1. Now,
(16)hκd2∘2l2=∨κphp=d2∘2l2≥∨κd1∘1l1d1,l1∈D1,hd1=d2,hl1=l2≥∨κd1∧κl1d1,l1∈D1,hd1=d2,hl1=l2=∨κd1d1∈D1,hd1=d2∧∨κl1l1∈D1,hl1=l2=hκd2∧hκl2.
Thereforehκd2∘2l2≥hκd2∧hκl2 for all d2 and l2∈D2.
Letl2∈L2 and d2∈D2/L2.
Sinceh is a homomorphism, hL1⊆L2. Now,
(17)hκl2=∨κl1l1∈D1,hl1=l2≥∨κd1d1∈D1,hd1=d2=hκd2.
Therefore,hκl2≥hκd2 for all l2∈L2 and d2∈D2/L2.
Hence,D2,hκ forms a fuzzy sub e-group of D2,∘2,L2.Theorem 31.
LetD1,∘1,L1 and D2,∘2,L2 be the two e-groups. Let h be a bijective homomorphism from D1,∘1,L1 to D2,∘2,L2 and D2,κ be a normal fuzzy sub e-group of D2,∘2,L2. Then, D1,h−1κ forms a normal fuzzy sub e-group of D1,∘1,L1.Proof.
From Theorem29, we can say that D1,h−1κ forms a fuzzy sub e-group of D1,∘1,L1.
SinceD2,κ forms a normal fuzzy sub e-group of D2,∘2,L2,κd2∘2l2=κl2∘2d2 for all d2, l2∈D2.
Letd1 and l1 be the two elements of D1. Then,
(18)h−1κd1∘1l1=κhd1∘1l1=κhd1∘2hl1=κhl1∘2hd1=κhl1∘1d1=h−1κl1∘1d1.
Therefore,h−1κd1∘1l1=h−1κl1∘1d1 for all d1 and l1∈D1.
Hence,D1,h−1κ forms a normal fuzzy sub e-group of D1,∘1,L1.Theorem 32.
LetD1,∘1,L1 and D2,∘2,L2 be the two e-groups. Let h be a bijective homomorphism from D1,∘1,L1 to D2,∘2,L2 and D1,κ be a normal fuzzy sub e-group of D1,∘1,L1. Then, D2,hκ forms a normal fuzzy sub e-group of D2,∘2,L2.Proof.
From Theorem30, we can say that D2,hκ is a fuzzy sub e-group of D2,∘2,L2.
SinceD1,κ forms a normal fuzzy sub e-group of D1,∘1,L1, κd1∘1l1=κl1∘1d1 for all d1, l1∈D1.
Letd2 and l2 be the two elements of D2.
Suppose that there are uniqued1 and l1∈D1, such that d2=hd1 and l2=hl1. Now,
(19)hκd2∘2l2=∨κphp=d2∘2l2=∨κd1∘1l1d1,l1∈D1,hd1=d2,hl1=l2∨κl1∘1d1d1,l1∈D1,hd1=d2,hl1=l2=∨κphp=l2∘2d2=hκl2∘2d2.
Therefore,hκd2∘2l2=hκl2∘2d2∀d2,l2∈D2.
Hence,D2,hκ forms a normal fuzzy sub e-group of D2,∘2,L2.


## 6. Conclusion

In this paper, we presented a brief demonstration of fuzzy sub e-groups and its properties. A condition is given for a FS of an e-group to be a fuzzy sub e-group. We have demonstrated that any fuzzy sub e-group forms a fuzzy subgroup. However, the reverse is not always true. Therefore, fuzzy sub e-group is the generalization of fuzzy subgroup. We have presented the difference between FSG and fuzzy sub e-group. We have discussed about the idea of normal fuzzy sub e-groups and level fuzzy e-subgroups. Finally, we have explained the effect of e-group homomorphism on fuzzy sub e-groups. In future, we will work on important theorems like Lagrange’s theorem and Sylow theorem in fuzzy sub e-groups.



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