Ratio Mathematica Volume 38, 2020, pp. 237-259

Some characterizations of fuzzy comultisets
and quotient fuzzy multigroups

Paul Augustine Ejegwa∗

Abstract

The idea of fuzzy multisets has been applied to some group theoretic
notions. Nonetheless, the notions of cosets and quotient groups have
not been substantiated in fuzzy multigroup environment. The aim of
this paper is to present the concepts of cosets and quotient groups in
fuzzy multigroup context with some related results. To start with, the
connection between fuzzy comultisets of fuzzy multigroups and the
cosets of groups is established. Some characterizations of fuzzy co-
multisets are outlined with proofs. In addition, quotient fuzzy multi-
group is proposed and some of its properties are explored. It is proven
that a normal fuzzy submultigroup, H̃ of a fuzzy multigroup, G̃ is
commutative if and only if the quotient fuzzy multigroup, G̃

H̃
of G̃ by

H̃ is commutative. Finally, group theoretic isomorphism theorems
are established in fuzzy multigroup setting.

Keywords: fuzzy multiset; fuzzy multigroup; fuzzy comultiset; quo-
tient/factor fuzzy multigroup.

2010 AMS subject classifications: 03E72, 08A72, 20N25. 1

∗Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B.
2373, Makurdi, Nigeria; ocholohi@gmail.com; ejegwa.augustine@uam.edu.ng

1Received on February 11th, 2020. Accepted on April 26th, 2020. Published on June 30th,
2020. doi: 10.23755/rm.v38i0.501. ISSN: 1592-7415. eISSN: 2282-8214. c©P. A. Ejegwa
This paper is published under the CC-BY licence agreement.

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Paul Augustine Ejegwa

1 Introduction
Fuzzy set theory proposed in 1965 by Zadeh (1965), although with vehement

opposition as at then, has been extensively researched with applicative expressions
ranging from engineering and computer science to medical diagnosis and social
behavior, etc. In a way of extending the application of fuzzy sets to group theory,
Rosenfeld (1971) proposed the notion of fuzzy groups as an extension of group
theory and some number of results were obtained. Several studies have been car-
ried out on some group theoretic notions in fuzzy group setting (see Ajmal and
Prajapati, 1992; Bhattacharya and Mukherjee, 1987; Ejegwa and Otuwe, 2019;
Mukherjee and Bhattacharya, 1986, 1984; Onasanya and Ilori, 2013).

With the interest derived from fuzzy sets and multisets (see Blizard, 1989),
the idea of fuzzy multisets or fuzzy bags was proposed in (Yager, 1986) as a gen-
eralization of fuzzy sets in multiset framework. Myriad of works have been car-
ried out on the fundamentals and properties of fuzzy multisets (see Biswas, 1999;
Ejegwa, 2014, 2019a; Miyamoto, 1996; Miyamoto and Mizutani, 2004; Onasanya
and Sholabomi, 2019). In recent times, the concept of fuzzy multigroups was in-
troduced as an application of fuzzy multisets to group theory (Shinoj et al., 2015).
The ideas of abelian fuzzy multigroups and order of fuzzy multigroups have been
studied with some results (Baby et al., 2015; Ejegwa, 2018b), and the notions
of center and centralizer in fuzzy multigroup context were established (Ejegwa,
2018b). In the same vein, the concept of fuzzy multigroupoids was introduced and
the idea of fuzzy submultigroups was explored with a number of results (Ejegwa,
2018d). The concepts of normal subgroups, characteristic subgroups and Frattini
subgroups have been established in fuzzy multigroup setting with some results
(Ejegwa, 2018a; Ejegwa et al., 2020; Ejegwa, 2020c). In (Ejegwa, 2018c), the idea
of homomorphism in the environment of fuzzy multigroups was defined and some
homomorphic properties of fuzzy multigroups were elaborated. Subsequently, the
idea of direct product of fuzzy multigroups was proposed and a number of results
were established (Ejegwa, 2019b). The idea of alpha-cuts of fuzzy multigroups
and its homomorphic properties have been studied (Ejegwa, 2020b,a). The con-
cept of fuzzy multigroups was redefined by Rasuli (2020) as an extension of the
work in (Anthony and Sherwood, 1979).

The present paper is a further study of fuzzy multigroups in group theoretic
analogs. Motivated by the researches done in fuzzy multigroups so far, it is expe-
dient to investigate the notions of cosets and quotient groups in the light of fuzzy
multigroups to strengthen the theory of fuzzy multigroups. Establishing the ideas
of fuzzy comultisets and quotient/factor multigroups shall enhance the plausibil-
ity of studying nilpotency and solvability in fuzzy multigroup setting. Actually,
this work is an application of fuzzy multisets to cosets and factor groups. In so
doing, this paper assay to introduce fuzzy comultisets and quotient fuzzy multi-

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Fuzzy comultisets and quotient fuzzy multigroups

groups with some analog results. The relationship between fuzzy comultisets of
fuzzy multigroups and that of cosets of groups is examined, and the isomorphism
theorems are duly established. By organization, the paper is thus presented: Sec-
tion 2 provides some preliminaries on fuzzy multisets and fuzzy multigroups. In
Section 3, the idea of fuzzy comultisets is proposed and some of its properties are
discussed. Section 4 discusses the concept of quotient or factor fuzzy multigroups
with some results. Finally, Section 5 concludes the paper and provides direction
for future studies.

2 Preliminaries
In this section, we present some existing definitions and results to be used in

the sequel.

2.1 Fuzzy multisets
Definition 2.1. (Yager, 1986) Assume X is a set of elements. Then, a fuzzy
bag/multiset, G̃ drawn from X can be characterized by a count membership func-
tion, CMG̃ such that

CMG̃ : X → Q,

where Q is the set of all crisp bags or multisets from the unit interval, I = [0,1].
A fuzzy multiset, G̃ can be characterized by a function

CMG̃ : X → N
I or CMG̃ : X → [0,1] → N,

where I = [0,1] and N = N∪{0}.
By Miyamoto and Mizutani (2004), it implies that CMG̃(x) for x ∈ X is

given as
CMG̃(x) = {µ

1
G̃
(x),µ2

G̃
(x), ...,µn

G̃
(x), ...},

where µ1
G̃
(x),µ2

G̃
(x), ...,µn

G̃
(x), ... ∈ [0,1] such that µ1

G̃
(x) ≥ µ2

G̃
(x) ≥ ... ≥

µn
G̃
(x) ≥ ..., whereas in a finite case, we write

CMG̃(x) = {µ
1
G̃
(x),µ2

G̃
(x), ...,µn

G̃
(x)},

for µ1
G̃
(x) ≥ µ2

G̃
(x) ≥ ... ≥ µn

G̃
(x).

A fuzzy multiset, G̃ can be represented in the form G̃ = {
〈CMG̃(x)〉

x
| x ∈

X}.
We denote the set of all fuzzy multisets by FMS(X).

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Paul Augustine Ejegwa

Definition 2.2. (Yager, 1986) Let G̃,H̃ ∈ FMS(X). Then, H̃ is called a fuzzy
submultiset of G̃ written as H̃ ⊆ G̃ if CMH̃(x) ≤ CMG̃(x), ∀x ∈ X. Also, if
H̃ ⊆ G̃ and H̃ 6= G̃, then H̃ is called a proper fuzzy submultiset of G̃ and denoted
as H̃ ⊂ G̃.

Definition 2.3. (Yager, 1986) Let G̃,H̃ ∈ FMS(X). Then, G̃ and H̃ are com-
parable to each other if and only if H̃ ⊆ G̃ or G̃ ⊆ H̃, and G̃ = H̃ if and only if
CMG̃(x) = CMH̃(x), ∀x ∈ X.

Definition 2.4. (Miyamoto, 1996) Let G̃,H̃ ∈ FMS(X). Then, the intersection
and union of G̃ and H̃, denoted by G̃∩H̃ and G̃∪H̃ are defined by the rules that
for any object x ∈ X,

(i) CMG̃∩H̃(x) = CMG̃(x)∧CMH̃(x),

(ii) CMG̃∪H̃(x) = CMG̃(x)∨CMH̃(x),

where ∧ and ∨ denote minimum and maximum operations, respectively.

Before finding the intersection and union of G̃ and H̃, the membership sequences
of G̃ and H̃ should be equal. If not, it could be completed by affixing zero(s).

2.2 Fuzzy multigroups
We denote group by X and assume that all fuzzy multigroups are drawn from

FMG(X), which is the set of all fuzzy multigroups of X.

Definition 2.5. (Shinoj et al., 2015) A fuzzy multiset G̃ of X is said to be a fuzzy
multigroup of X if it satisfies the following two conditions:

(i) CMG̃(xy) ≥ CMG̃(x)∧CMG̃(y), ∀x,y ∈ X,

(ii) CMG̃(x
−1) = CMG̃(x), ∀x ∈ X.

It can be easily verified that if G̃ is a fuzzy multigroup of X, then

CMG̃(e) =
∨
x∈X

CMG̃(x),

that is, CMG̃(e) is the tip of G̃.

Remark 2.1. (Shinoj et al., 2015) We notice that a fuzzy multiset, G̃ of a group
X is a fuzzy multigroup if ∀x,y ∈ X,

CMG̃(xy
−1) ≥ CMG̃(x)∧CMG̃(y)

holds.

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Fuzzy comultisets and quotient fuzzy multigroups

Definition 2.6. (Shinoj et al., 2015) Let G̃ be a fuzzy multigroup of a group X.
Then G̃−1 is defined by CMG̃−1(x) = CMG̃(x

−1), ∀x ∈ X.
By Definition 2.5, we get CMG̃−1(x) = CMG̃(x

−1) = CMG̃(x). That is,
G̃−1 = G̃. Thus, G̃ ∈ FMG(X) ⇔ G̃−1 ∈ FMG(X).

Proposition 2.1. (Shinoj et al., 2015) Let G̃,H̃ ∈ FMG(X). Then, G̃ ∩ H̃ ∈
FMG(X).

Definition 2.7. (Ejegwa, 2018d) Let {G̃i}i∈I,I = 1, ...,n be an arbitrary family
of fuzzy multigroups of X. Then,

CM⋂
i∈I G̃i

(x) =
∧
i∈I

CMG̃i(x), ∀x ∈ X

and
CM⋃

i∈I G̃i
(x) =

∨
i∈I

CMG̃i(x), ∀x ∈ X.

The family of fuzzy multigroups {G̃i}i∈I of X is said to have inf or sup assuming
chain if either G̃1 ⊆ G̃2 ⊆ ... ⊆ G̃n or G̃1 ⊇ G̃2 ⊇ ... ⊇ G̃n, respectively.

Definition 2.8. (Baby et al., 2015) Let G̃ ∈ FMG(X). Then, G̃ is said to be
commutative if for all x,y ∈ X,

CMG̃(xy) = CMG̃(yx).

Definition 2.9. (Ejegwa, 2018a) Let G̃,H̃ ∈ FMG(X). Then, the product, G̃◦H̃
of G̃ and H̃ is defined to be a fuzzy multiset of X as follows:

CMG̃◦H̃(x) =

{ ∨
x=yz[CMG̃(y)∧CMH̃(z)], if ∃y,z ∈ X such that x = yz

0, otherwise.

Definition 2.10. (Ejegwa, 2018d) Let G̃ ∈ FMG(X). A fuzzy submultiset, H̃
of G̃ is called a fuzzy submultigroup of G̃ denoted by H̃ ⊆ G̃ if H̃ is a fuzzy
multigroup. A fuzzy submultigroup, H̃ of G̃ is a proper fuzzy submultigroup
denoted by H̃ ⊂ G̃, if H̃ ⊆ G̃ and H̃ 6= G̃.

Definition 2.11. (Ejegwa, 2018a) Let H̃,G̃ ∈ FMG(X) such that H̃ ⊆ G̃. Then,
H̃ is called a normal fuzzy submultigroup of G̃ if

CMH̃(xyx
−1) = CMH̃(y), ∀x,y ∈ X.

Definition 2.12. (Ejegwa, 2018a) Let H̃ be a fuzzy submultiset of G̃ ∈ FMG(X).
Then, the normalizer of H̃ in G̃ is the set given by

N(H̃) = {g ∈ X | CMH̃(gy) = CMH̃(yg), ∀y ∈ X}.

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Paul Augustine Ejegwa

Theorem 2.1. (Ejegwa, 2018a) Let X be a finite group and H̃ be a fuzzy sub-
multigroup of G̃ ∈ FMG(X). Define

H = {g ∈ X | CMH̃(g) = CMH̃(e)},
K = {x ∈ X | CMH̃x(y) = CMH̃e(y)},

where e denotes the identity element of X. Then H and K are subgroups of X.
Again, H = K.

Definition 2.13. (Ejegwa, 2018d; Shinoj et al., 2015) Let G̃ ∈ FMG(X). Then,
the set G̃∗ defined by

G̃∗ = {x ∈ X | CMG̃(x) > 0}
is the level set or support of G̃. It follows that G̃∗ is a subgroup of X.

Also, the set G̃∗ defined by

G̃∗ = {x ∈ X | CMG̃(x) = CMG̃(e)}
is a subgroup of X.

Definition 2.14. (Ejegwa, 2018c) Let X and Y be groups and let f : X → Y
be a homomorphism. Suppose G̃ and H̃ are fuzzy multigroups of X and Y ,
respectively. Then, f induces a homomorphism from G̃ to H̃ which satisfies

(i) CMG̃(f
−1(y1y2)) ≥ CMG̃(f

−1(y1))∧CMG̃(f
−1(y2)), ∀y1,y2 ∈ Y ,

(ii) CMH̃(f(x1x2)) ≥ CMH̃(f(x1))∧CMH̃(f(x2)), ∀x1,x2 ∈ X,
where

(i) the image of G̃ under f, denoted by f(G̃), is a fuzzy multiset over Y defined
by

CMf(G̃)(y) =

{ ∨
x∈f−1(y) CMG̃(x), f

−1(y) 6= ∅
0, otherwise

for each y ∈ Y .

(ii) the inverse image of H̃ under f, denoted by f−1(H̃), is a fuzzy multiset
over X defined by

CMf−1(H̃)(x) = CMH̃(f(x)), ∀x ∈ X.

Proposition 2.2. (Ejegwa, 2018c) Let f : X → Y be a homomorphism and
G̃ ∈ FMG(X). If f is injective, then f−1(f(G̃)) = G̃.
Theorem 2.2. (Ejegwa, 2018c) Let X and Y be groups and f : X → Y be an
isomorphism. Then the following statements hold.

(i) G̃ ∈ FMG(X) if and only if f(G̃) ∈ FMG(Y ).

(ii) H̃ ∈ FMG(Y ) if and only if f−1(H̃) ∈ FMG(X).

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Fuzzy comultisets and quotient fuzzy multigroups

3 Fuzzy comultiset and some of its properties
In this section, we define fuzzy comultiset and characterize some of its prop-

erties.

Definition 3.1. Suppose H̃ is a fuzzy submultigroup of a fuzzy multigroup G̃ of
X. Then, the fuzzy submultiset, yH̃ of G̃ for y ∈ X defined by

CMyH̃(x) = CMH̃(y
−1x), ∀x ∈ X

is called the left fuzzy comultiset of H̃. Similarly, the fuzzy submultiset, H̃y of G̃
for y ∈ X defined by

CMH̃y(x) = CMH̃(xy
−1), ∀x ∈ X

is called the right fuzzy comultiset of H̃.

The following result proves that the right and left fuzzy comultisets of a fuzzy
submultigroup in a fuzzy multigroup are equal.

Proposition 3.1. If H̃ is a fuzzy submultigroup of G̃ ∈ FMG(X), then the right
and left fuzzy comultisets of H̃ in G̃ are identical.

Proof. Let x,y,∈ X. Assume H̃ is a fuzzy submultigroup of G̃. Then, we have

CMyH̃(x) = CMH̃(y
−1x) ≥ CMH̃(y)∧CMH̃(x)

= CMH̃(x)∧CMH̃(y)
= CMH̃(x)∧CMH̃(y

−1).

Suppose by hypothesis, CMH̃(x)∧CMH̃(y
−1) = CMH̃(xy

−1). Then, we have

CMyH̃(x) ≥ CMH̃y(x).

Again,

CMH̃y(x) = CMH̃(xy
−1) ≥ CMH̃(x)∧CMH̃(y)

= CMH̃(y)∧CMH̃(x)
= CMH̃(y

−1)∧CMH̃(x).

By the same hypothesis, we get

CMH̃y(x) ≥ CMyH̃(x).

Hence, CMyH̃(x) = CMH̃y(x) ⇒ yH̃ = H̃y.

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Paul Augustine Ejegwa

Remark 3.1. Let H̃ be a fuzzy submultigroup of G̃ ∈ FMG(X). We notice that

(i) the right and left fuzzy comultisets of H̃ in G̃ are fuzzy submultigroups of
G̃.

(ii) xH̃ = yH̃ = zH̃ = H̃, ∀x,y,z ∈ X. This is not applicable in the conven-
tional case.

(iii) there is a one-to-one correspondence between the set of right fuzzy comul-
tisets and the set of left fuzzy comultisets of H̃ in G̃.

(iv) the number of fuzzy comultisets of H̃ in G̃ equals the cardinality of H̃∗.

(v) xH̃ ∩yH̃ ∩zH̃ = H̃ = xH̃ ∪yH̃ ∪zH̃, ∀x,y,z ∈ X.

Theorem 3.1. Let H̃ be a fuzzy submultigroup of G̃ ∈ FMG(X). Then, gH̃ =
hH̃ for g,h ∈ X if and only if

CMH̃(g
−1h) = CMH̃(h

−1g) = CMH̃(e).

Proof. Let gH̃ = hH̃. Then, CMgH̃(g) = CMhH̃(g) and CMgH̃(h) = CMhH̃(h)
∀g,h ∈ X. Hence,

CMH̃(g
−1h) = CMH̃(h

−1g) = CMH̃(e).

Conversely, let CMH̃(g
−1h) = CMH̃(h

−1g) ∀g,h ∈ X. For every x ∈ X,
we have

CMgH̃(x) = CMH̃(g
−1x) = CMH̃(g

−1hh−1x)

≥ CMH̃(g
−1h)∧CMH̃(h

−1x)

= CMH̃(h
−1x)

= CMhH̃(x).

Similarly,

CMhH̃(x) = CMH̃(h
−1x) = CMH̃(h

−1gg−1x)

≥ CMH̃(h
−1g)∧CMH̃(g

−1x)

= CMH̃(g
−1x)

= CMgH̃(x).

Hence, CMgH̃(x) = CMhH̃(x) ⇒ gH̃ = hH̃.

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Fuzzy comultisets and quotient fuzzy multigroups

Corolary 3.1. Let H̃ be a fuzzy submultigroup of G̃ ∈ FMG(X). Then H̃g =
H̃h for g,h ∈ X if and only if

CMH̃(gh
−1) = CMH̃(hg

−1) = CMH̃(e).

Proof. Straightforward from Theorem 3.1.

Proposition 3.2. Let H̃,G̃ ∈ FMG(X) such that H̃ ⊆ G̃. If gH̃ = hH̃, then
CMH̃(g) = CMH̃(h), ∀g,h ∈ X.

Proof. Let g,h ∈ X. Suppose gH̃ = hH̃, then we have

CMgH̃(g) = CMhH̃(g) ⇒ CMH̃(g
−1g) = CMH̃(h

−1g)

⇒ CMH̃(e) = CMH̃(h
−1g)

∀g,h ∈ X. The fact that,

CMH̃(e) = CMH̃(h
−1g) ⇒ CMH̃(h) = CMH̃(g),

the result follows.
Alternatively, suppose z ∈ X, we get

CMgH̃(z) = CMhH̃(z) ⇒ CMH̃(g
−1z) = CMH̃(h

−1z)

⇒ CMH̃z−1(g) = CMH̃z−1(h)
⇒ CMH̃(g) = CMH̃(h),

because H̃z−1 = H̃.

Theorem 3.2. Let G̃ ∈ FMG(X). Any fuzzy submultigroup H̃ of G̃ and for any
z ∈ X, the fuzzy submultiset, zH̃z−1, where CMzH̃z−1(x) = CMH̃(z

−1xz) for
each x ∈ X is a fuzzy submultigroup of G̃.

Proof. Let x,y ∈ X and H̃ ⊆ G̃. We prove that zH̃z−1 is a fuzzy submultigroup
of G̃. Now

CMzH̃z−1(xy
−1) = CMH̃(z

−1xy−1z)

= CMH̃(z
−1xzz−1y−1z)

≥ CMH̃(z
−1xz)∧CMH̃(z

−1y−1z)

= CMzH̃z−1(x)∧CMzH̃z−1(y
−1)

= CMzH̃z−1(x)∧CMzH̃z−1(y), ∀z ∈ X.

Hence, zH̃z−1 is a fuzzy submultigroup of G̃.

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Paul Augustine Ejegwa

Corolary 3.2. Let {H̃i}i∈I ∈ FMG(X), then

(i)
⋂

i∈I zH̃iz
−1 ∈ FMG(X), ∀z ∈ X,

(ii)
⋃

i∈I zH̃iz
−1 ∈ FMG(X), ∀z ∈ X provided {H̃i}i∈I have sup/inf assum-

ing chain.

Proof. The results follow from Theorem 3.2.

The following results are the application of product of fuzzy multigroups to
the idea of fuzzy comultisets.

Proposition 3.3. Suppose H̃ is a fuzzy submultigroup of H̃ ∈ FMG(X). Then;

(i) H̃g ◦ H̃g = H̃g.

(ii) H̃g ◦ H̃h = H̃h◦ H̃g.

(iii) (H̃g ◦ H̃h)−1 = (H̃h)−1 ◦ (H̃g)−1.

(iv) (H̃g ◦ H̃h)−1 = H̃g ◦ H̃h.

Proof. Using Definition 2.9, the results follow.

Remark 3.2. Proposition 3.3 also holds for left fuzzy comultisets.

Proposition 3.4. If H̃ is a fuzzy submultigroup of a commutative fuzzy multigroup
G̃ of X, then

(i) H̃y ◦ H̃z = H̃yz, ∀y,z ∈ X,

(ii) yH̃ ◦zH̃ = yzH̃, ∀y,z ∈ X.

Proof. Let H̃ ∈ FMG(X) and x,y,z ∈ X, then we have

(i)

CMH̃y◦H̃z(x) =
∨
x=zy

[CMH̃y(z)∧CMH̃z(y)], ∀y,z ∈ X

=
∨
x=zy

[CMH̃(zy
−1)∧CMH̃(yz

−1)], ∀y,z ∈ X

=
∨
x=zy

[CMH̃∩H̃((zy
−1)(yz−1))], ∀y,z ∈ X

= CMH̃(xz
−1y−1)

= CMH̃yz(x).

Hence, H̃y ◦ H̃z = H̃yz.

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Fuzzy comultisets and quotient fuzzy multigroups

(ii) Similar to (i).

Corolary 3.3. Suppose H̃ is a fuzzy submultigroup of a commutative fuzzy multi-
group G̃ of X. Then, the following statements are equivalent.

(i) (H̃y ◦ H̃z)−1 = H̃y ◦ H̃z.

(ii) H̃y ◦ H̃z = H̃yz.

Proof. The result is easy to see by combining Definition 2.9 and Proposition 3.4.

Remark 3.3. If (H̃y ◦ H̃y)−1 = H̃y ◦ H̃z and H̃y ◦ H̃z = H̃yz, then (H̃y ◦
H̃z)−1 = H̃yz.

Theorem 3.3. If H̃ is a fuzzy submultigroup of G̃ ∈ FMG(X) such that G̃ is
commutative, then H̃g ◦ H̃h = H̃gh if and only if gH̃ ◦hH̃ = ghH̃, ∀ g,h ∈ X.
Consequently, H̃gh = ghH̃.

Proof. Suppose H̃g ◦ H̃h = H̃gh. By Definition 2.9, we get

CMH̃gh(x) = CMH̃g◦H̃h(x)

=
∨
y∈X

[CMH̃g(y)∧CMH̃h(y
−1x)]

=
∨
y∈X

[CMH̃(yg
−1)∧CMH̃(y

−1xh−1)]

=
∨
y∈X

[CMH̃(g
−1y)∧CMH̃(h

−1y−1x)]

=
∨
y∈X

[CMgH̃(y)∧CMhH̃(y
−1x)]

= CMgH̃◦hH̃(x)

= CMghH̃(x)

⇒ gH̃ ◦hH̃ = ghH̃.

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Paul Augustine Ejegwa

Conversely, assuming gH̃ ◦hH̃ = ghH̃. Then

CMghH̃(x) = CMgH̃◦hH̃(x)

=
∨
y∈X

[CMgH̃(y)∧CMhH̃(y
−1x)]

=
∨
y∈X

[CMH̃(g
−1y)∧CMH̃(h

−1y−1x)]

=
∨
y∈X

[CMH̃(yg
−1)∧CMH̃(y

−1xh−1)]

=
∨
y∈X

[CMH̃g(y)∧CMH̃h(y
−1x)]

= CMH̃g◦H̃h(x)

= CMH̃gh(x)

⇒ H̃g ◦ H̃h = H̃gh. Hence, the result follow.
Theorem 3.4. Suppose G̃ ∈ FMG(X) and H̃ a fuzzy submultigroup of G̃. Define

H = {g ∈ X | CMH̃(g) = CMH̃(e)}.

Then Hx = Hy ⇔ H̃x = H̃y, ∀x,y ∈ X. Similarly, xH = yH ⇔ xH̃ = yH̃.
Proof. This result gives a relationship between fuzzy comultisets of a fuzzy sub-
multigroup of a fuzzy multigroup and the cosets of a subgroup of a given group.

By Theorem 2.1, we know that H is a subgroup of X and

H = {x ∈ X | CMH̃x(z) = CMH̃e(z)}.

Now, suppose that Hx = Hy. Then xy−1 ∈ H. Thus

CMH̃xy−1(z) = CMH̃e(z)∀z ∈ X and so CMH̃(zyx
−1) = CMH̃(z).

Put z = zy−1, we get

CMH̃(zy
−1yx−1) = CMH̃(zy

−1) ⇒ CMH̃(zx
−1) = CMH̃(zy

−1)

⇒ CMH̃x(z) = CMH̃y(z)

and so, H̃x = H̃y.
Conversely, suppose that H̃x = H̃y, that is CMH̃x(z) = CMH̃y(z), ∀z ∈ X.

This implies that
CMH̃(zx

−1) = CMH̃(zy
−1).

Put z = y, we get
CMH̃(yx

−1) = CMH̃(e).

So, yx−1 ∈ H. Thus, Hx = Hy.
The proof of xH = yH ⇔ xH̃ = yH̃ is similar.

248



Fuzzy comultisets and quotient fuzzy multigroups

4 Quotient fuzzy multigroups
In this section, we present the notion of quotient groups in fuzzy multigroup

setting and establish the isomorphism theorems.

Definition 4.1. Suppose G̃ is a fuzzy multigroup of X and H̃ a normal fuzzy
submultigroup of G̃. Then, the union of the set of left/right fuzzy comultisets of
H̃ such that the fuzzy comultisets satisfy

xH̃ ◦yH̃ = xyH̃, ∀x,y ∈ X

is called quotient or factor fuzzy multigroup of G̃ by H̃, denoted by G̃
H̃

.

Remark 4.1. Suppose G̃
H̃

is a factor fuzzy multigroup of G̃ by H̃, it implies that

H̃ is a normal fuzzy submultigroup of G̃ and G̃
H̃

= eH̃ = H̃. This property is not
applicable in classical case.

Remark 4.2. Suppose G̃ is a fuzzy multigroup of X, and H̃ a normal fuzzy sub-
multigroup of G̃. Then

(i) if Ĩ is a fuzzy submultigroup of G̃ such that H̃ ⊆ Ĩ ⊆ G̃, then Ĩ
H̃

is a fuzzy

submultigroup of G̃
H̃

.

(ii) every fuzzy submultigroup of G̃
H̃

is of the form Ĩ
H̃

, for some fuzzy submulti-
group Ĩ of G̃ such that H̃ ⊆ Ĩ ⊆ G̃.

Theorem 4.1. If H̃ is a normal fuzzy submultigroup of G̃ ∈ FMG(X). Then H̃
is commutative if and only if G̃

H̃
is commutative.

Proof. Let x,y ∈ X. Suppose H̃ is commutative, then

CMH̃(xyx
−1y−1) = CMH̃(e),

and hence,
CMH̃(xy) = CMH̃(yx).

Consequently, H̃ is a normal fuzzy submultigroup of G̃ by Definition 2.11.
Thus, since

CMH̃(xy(yx)
−1) = CMH̃(xyx

−1y−1) = CMH̃(e),

we have

CMH̃(xy(yx)
−1) = CMH̃(e) ⇒ CMH̃(xy(yx)

−1) = CMH̃(xy(xy)
−1)

⇒ CMH̃yx(xy) = CMH̃xy(xy).

249



Paul Augustine Ejegwa

Thus, H̃xy = H̃yx. It follows that, H̃x◦H̃y = H̃y◦H̃x since H̃x◦H̃y = H̃xy
and H̃y ◦ H̃x = H̃yx by Proposition 3.4. Hence, G̃

H̃
is commutative.

Conversely, assume G̃
H̃

is commutative, then

H̃x◦ H̃y = H̃y ◦ H̃x ⇒ H̃xy = H̃yx.

Thus,
CMH̃(xy(yx)

−1) = CMH̃(e) ⇒ CMH̃(xy) = CMH̃(yx),
completes the proof.

Theorem 4.2. Suppose G̃ ∈ FMG(X) and H̃, Ĩ are normal fuzzy submultigroups
of G̃ and H̃ ⊆ Ĩ, then Ĩ

H̃
is a normal fuzzy submultigroup of G̃

H̃
.

Proof. Let x ∈ X. Then CM Ĩ
H̃

(x) ≤ CM G̃
H̃

(x) since H̃ ⊆ Ĩ and, H̃ and Ĩ

are normal fuzzy submultigroups of G̃. So, Ĩ
H̃

is a fuzzy submultigroup of G̃
H̃

.
Subsequently,

CM Ĩ
H̃

(yxy−1) = CM Ĩ
H̃

(x)∀x,y ∈ X.

Hence, Ĩ
H̃

is a normal fuzzy submultigroup of G̃
H̃

by Definition 2.11.

Remark 4.3. Let G̃ be a fuzzy multigroup of X, and Ĩ a normal fuzzy submulti-
group of G̃. Then, every normal fuzzy submultigroup of G̃

H̃
is of the form Ĩ

H̃
, for

some normal fuzzy submultigroup H̃ of G̃ such that H̃ ⊆ Ĩ ⊆ G̃.

Theorem 4.3. Suppose G̃,H̃ ∈ FMG(X) and H̃ a normal fuzzy submultigroup
of G̃. Then H̃∩G̃

H̃∗
is a normal fuzzy submultigroup of H̃.

Proof. By Definition 2.13, H̃∗ is a subgroup of X and H̃ ∩ G̃ ∈ FMG(X) by
Proposition 2.1. So, H̃∩G̃

H̃∗
is a fuzzy multigroup of X. Since H̃ is a normal fuzzy

submultigroup of G̃, then H̃∩G̃ is a fuzzy submultigroup of G̃ and H̃∩G̃
H̃∗

is a fuzzy

submultigroup of H̃. We show that H̃∩G̃
H̃∗

is a normal fuzzy submultigroup of H̃.

Let x,y ∈ H̃∗. Then xyx−1 ∈ H̃∗ since

CMH̃(xyx
−1) = CMH̃(y) > 0

by definition of H̃∗. This proves that H̃∗ is a normal subgroup of X.
It is easy to see that H̃ ∩ G̃ is normal since

CMH̃∩G̃(xyx
−1) = CMH̃(xyx

−1)∧CMG̃(xyx
−1)

= CMH̃(y)∧CMG̃(y)
= CMH̃∩G̃(y).

250



Fuzzy comultisets and quotient fuzzy multigroups

In fact, H̃ ∩ G̃ is a normal fuzzy submultigroup since H̃ ∩ G̃ = H̃, and H̃ is a
normal fuzzy submultigroup of G̃. Hence, H̃∩G̃

H̃∗
is a normal fuzzy submultigroup

of H̃.

Theorem 4.4. Suppose G̃ is a fuzzy multigroup of X, H̃ ⊆ G̃ and N(H̃) is a nor-
malizer. Then N(H̃) is a subgroup of X and H̃

N(H̃)
is a normal fuzzy submultigroup

of G̃.

Proof. Clearly, e ∈ N(H̃). Let x,y ∈ N(H̃). Then for any z ∈ X, we have

CMH̃((xy
−1)z) = CMH̃(x(y

−1z))

= CMH̃((y
−1z)x)

= CMH̃(y
−1(zx))

= CMH̃(y(zx)
−1)

= CMH̃(y(x
−1z−1))

= CMH̃(z(xy
−1)).

Hence, xy−1 ∈ N(H̃). Therefore, N(H̃) is a subgroup of X. By Definition 4.1,
it follows that H̃

N(H̃)
∈ FMG(N(H̃)) and clearly, H̃

N(H̃)
is a fuzzy submultigroup

of H̃. Since, CM H̃
N(H̃)

(xyx−1) = CM H̃
N(H̃)

(y), ∀x,y ∈ X, it implies that H̃
N(H̃)

is

a normal fuzzy submultigroup of H̃.

Theorem 4.5. Suppose G̃ is a commutative fuzzy multigroup of X and H̃ a normal
fuzzy submultigroup of G̃. Then, there exists a natural homomorphism f : G̃ → G̃

H̃

defined by CMf(G̃)(y) = CMH̃(x
−1y), ∀x,y ∈ X.

Proof. Let f : G̃ → G̃
H̃

be a mapping defined by

CMf(G̃)(y) = CMH̃(x
−1y), ∀x,y ∈ X.

That is, CMf(G̃)(y) = CMxH̃(y) ⇒ f(G̃) = xH̃ (consequently, f(G̃∗) = xH̃∗).
Since f : G̃ → G̃

H̃
is derived from f : G̃∗ → G̃∗H̃∗ such that H̃∗ is a normal subgroup

of G̃∗, then to prove that f is a homomorphism, we show that

CMxyH̃(z) = CMxH̃◦yH̃(z), ∀z ∈ X ⇒ f(xy) = f(x)f(y).

Since H̃ is commutative, then

CMH̃(xz) = CMH̃(zx) ⇒ CMH̃(z
−1xz) = CMH̃(x), ∀z ∈ X.

251



Paul Augustine Ejegwa

It is certain that,

CMxH̃(z) = CMH̃(x
−1z) and CMyH̃(z) = CMH̃(y

−1z).

Then
CMxyH̃(z) = CMH̃((xy)

−1z).

Now,

CMxH̃◦yH̃(z) =
∨
z=rs

[CMxH̃(r)∧CMyH̃(s)]

=
∨
z=rs

[CMH̃(x
−1r)∧CMH̃(y

−1s)].

Similarly,

CMxyH̃(z) = CMH̃((xy)
−1z) = CMH̃(y

−1x−1z)

≥
∨
z=rs

[CMH̃(x
−1r)∧CMH̃(y

−1s)].

Suppose by hypothesis,

CMH̃(y
−1x−1z) =

∨
z=rs

[CMH̃(x
−1r)∧CMH̃(y

−1s)],

then it follows that CMxyH̃(z) = CMxH̃◦yH̃(z), ∀z ∈ X. Consequently, we have
f(xy) = f(x)f(y), ∀x,y ∈ X. Hence, f is a homomorphism.

Corolary 4.1. Let G̃,H̃ ∈ FMG(X) such that CMG̃(x) = CMG̃(y), ∀x,y ∈ X
and CMG̃(e) ≥ CMH̃(x), ∀x ∈ X. If f : G̃ →

G̃
H̃

is a natural homomorphism
defined by CMf(G̃)(y) = CMH̃(x

−1y), ∀x,y ∈ X, then f−1(f(H̃)) = G̃◦ H̃.

Proof. Let x ∈ X. To proof the result, we assume that f(x) = f(y), ∀x,y ∈ X.
Thus,

CMf−1(f(H̃))(x) =
∨
x∈X

[CMf(H̃)(f(x))]

=
∨
x∈X

[CMH̃(f
−1(f(y))]

= CMH̃(y).

252



Fuzzy comultisets and quotient fuzzy multigroups

Again,

CMG̃◦H̃(x) =
∨
x=zy

[CMG̃(z)∧CMH̃(y)]

=
∨
x∈X

[CMG̃(xy
−1)∧CMH̃(y)]

=
∨
x∈X

[CMG̃(e)∧CMH̃(y)]

= CMH̃(y).

Hence, the proof follows.

Remark 4.4. Assuming there is a bijective correspondence between every (nor-
mal) fuzzy submultigroup of G̃ that contains H̃ and the (normal) fuzzy submulti-
groups of G̃

H̃
. That is, if Ĩ is a (normal) fuzzy submultigroup of G̃ containing H̃,

then the corresponding (normal) fuzzy submultigroup of G̃
H̃

is f(Ĩ).

Theorem 4.6. Let X and Y be groups, f : X → Y be an isomorphism and H̃ a
normal fuzzy submultigroup of G̃ ∈ FMG(X) such that CMG̃(x) = CMG̃(y)∀x,y ∈
X with kerf = {e}. Then G̃

H̃
∼= f(G̃)

f(H̃)
.

Proof. By Theorem 2.2 and Definition 4.1, G̃
H̃

and f(G̃)
f(H̃)

are fuzzy multigroups,
respectively.

Let h : G̃
H̃
→ f(G̃)

f(H̃)
be defined by h(H̃x) = f(H̃)(f(x)), ∀x ∈ X. If H̃x =

H̃y, then CMH̃(xy
−1) = CMH̃(e). Since kerf = {e} meaning kerf ⊆ A

∗, then
f−1(f(H̃)) = H̃ by Proposition 2.2. Thus,

CMf−1(f(H̃))(xy
−1) = CMf−1(f(H̃))(e)

⇒
CMf(H̃)(f(xy

−1)) = CMf(H̃)(f(e))

⇒
CMf(H̃)(f(x)(f(y))

−1) = CMf(H̃)(f(e))

⇒
CMf(H̃)(f(x)) = CMf(H̃)(f(y)e

′) (where f(e) = e′).

Hence,

CMf(H̃)(f(x)) = CMf(H̃)(f(y)) ⇒ f(H̃)(f(x)) = f(H̃)(f(y)).

253



Paul Augustine Ejegwa

Hence, h is well-defined. It is also a homomorphism because

h(H̃xH̃y) = h(H̃xy) = f(H̃)(f(xy))

= f(H̃)(f(x)f(y))

= f(H̃)(f(x))f(H̃)(f(y))

= h(H̃x)h(H̃y).

Suppose f is an epimorphism, then ∃x ∈ X such that f(x) = y. Thus,

h(H̃x) = f(H̃)(f(x)) = f(H̃)(y).

Moreover,

f(H̃)(f(x)) = f(H̃)(f(y)) ⇒ CMf(H̃)(f(x)(f(y))
−1) = CMf(H̃)(e

′) ⇒

CMf(H̃)(f(xy
−1)) = CMf(H̃)(f(e)) ⇒ CMf−1(f(H̃))(xy

−1) = CMf−1(f(H̃))(e)

implies CMH̃(xy
−1) = CMH̃(e) ⇒ H̃x = H̃y, which proves that h is an iso-

morphism. Hence, the result follows.

Corolary 4.2. Let f : X → Y be an isomorphism and H̃ a normal fuzzy sub-
multigroup of G̃ ∈ FMG(Y ) such that CMG̃(x) = CMG̃(y), ∀x,y ∈ Y . Then
f(G̃)

f(H̃)
∼= G̃

H̃
.

Proof. By Theorem 2.2, f(G̃),f(H̃) ∈ FMG(X) and f(G̃)
f(H̃)

and f(G̃)
f(H̃)

are fuzzy
multigroups by Definition 4.1.

Again, since H̃ ∈ FMG(Y ), then

f(f−1(H̃)) = H̃.

If x ∈ kerf, then f(x) = e′ = f(e), and so

CMH̃(f(x)) = CMH̃(f(e)) ⇒ CMf−1(H̃)(x) = CMf−1(H̃)(e).

Hence, kerf ⊆ f−1(H̃∗). The proof is completed following the same logic as in
Theorem 4.6.

Theorem 4.7. Let H̃,G̃ ∈ FMG(X) and H̃ a normal fuzzy submultigroup of G̃.
Then G̃

G̃∗
≈ G̃

H̃
.

Proof. Let f be a natural homomorphism from G̃∗ onto
G̃∗
H̃∗

defined by f(xH̃∗) =

xH̃∗ ∀x ∈ G̃∗. Then, we have

CM
f( G̃

G̃∗
)
(xH̃∗) = ∨[CM G̃

G̃∗
(z)], ∀z ∈ G̃∗,f(z) = xH̃∗.

254



Fuzzy comultisets and quotient fuzzy multigroups

Since G̃
G̃∗

and G̃ are bijective correspondence to each other (by Remark 4.4) and

z = f−1(xH̃∗) = xH̃∗, it follows that

CM
f( G̃

G̃∗
)
(xH̃∗) = ∨[CM G̃

G̃∗
(z)], ∀z ∈ G̃∗,f(z) = xH̃∗

= ∨[CMG̃(y)], ∀y ∈ xH̃∗
= CM G̃

H̃

(xH̃∗), ∀x ∈ G̃∗,

because G̃
H̃

and G̃ are bijective correspondence to each other. Hence, it follows

that G̃
G̃∗
≈ G̃

H̃
.

Lemma 4.1. Suppose f : X → Y and G̃ ∈ FMG(X), then (f(G̃))∗ = f(G̃∗).
Proof. Straightforward.

Theorem 4.8. Let G̃ ∈ FMG(X). Suppose Y is a group and Ĩ ∈ FMG(Y )
such that G̃ ≈ Ĩ. Then, there exists a normal fuzzy submultigroup H̃ of G̃ such
that G̃

H̃
∼= Ĩ

Ĩ∗
.

Proof. Since G̃ ≈ Ĩ,∃ an epimorphism f of X onto Y such that f(G̃) = Ĩ.
Define H̃ ∈ FMG(X) as follows: ∀x ∈ X,

CMH̃(x) =

{
CMG̃(x) if x ∈ kerf
0, otherwise

Clearly, H̃ ⊆ G̃. If x ∈ kerf, then yxy−1 ∈ kerf, ∀y ∈ X, and so

CMH̃(yxy
−1) = CMG̃(yxy

−1) = CMG̃(x) = CMH̃(x), ∀y ∈ X.

If x /∈ kerf, then CMH̃(x) = 0 and so

CMH̃(yxy
−1) = CMH̃(x) = 0, ∀y ∈ X.

Hence, H̃ is a normal fuzzy submultigroup of G̃. Also, G̃ ≈ Ĩ ⇒ f(G̃) = Ĩ
which further implies (f(G̃))∗ = Ĩ∗ and f(G̃∗) = Ĩ∗ by Lemma 4.1. Let f = g.
Then g is a homomorphism of G̃∗ onto Ĩ∗ and ker g = H̃∗. Thus, there exists an
isomorphism h of G̃∗

H̃∗
onto Ĩ∗ such that h(xH̃∗) = g(x) = f(x), ∀x ∈ G̃∗. For

such an h, we have

CM
h( G̃

H̃
)
(z) = ∨[CM G̃

H̃

(xH̃∗)], ∀x ∈ G̃∗,h(xH̃∗) = z

= ∨(∨[CMG̃(y)], ∀y ∈ xH̃∗),∀x ∈ G̃∗,g(x) = z
= ∨[CMG̃(y)], ∀y ∈ G̃∗,g(y) = z
= ∨[CMG̃(y)], ∀y ∈ X,f(y) = z
= CMG̃(f

−1(z)) = CMf(G̃)(z) = CMĨ(z), ∀z ∈ Ĩ∗.

Therefore, G̃
H̃
∼= Ĩ

Ĩ∗
.

255



Paul Augustine Ejegwa

Theorem 4.9. Suppose G̃ is a fuzzy multigroup of X and H̃ a normal fuzzy sub-
multigroup of G̃. Then G̃

(H̃∩G̃)
' (H̃◦G̃)

H̃
.

Proof. From Definition 2.13, it is easy to infer that H̃∗ is also a normal subgroup
of X. By the Second Isomorphism Theorem for groups, we deduce

G̃∗

H̃∗ ∩ G̃∗
∼=
H̃∗G̃∗

H̃∗
.

Assume that
(H̃ ∩ G̃)∗ = H̃∗ ∩ G̃∗ and (H̃ ◦ G̃)∗ = H̃∗G̃∗.

Consequently, we have
G̃∗

(H̃ ∩ G̃)∗
∼=

(H̃ ◦ G̃)∗
H̃∗

,

where f is given by

f(x(H̃ ∩ G̃)∗) = xH̃∗, ∀x ∈ G̃∗.

Thus,

CM
f( G̃

H̃∩G̃
)
(yH̃∗) = C G̃

H̃∩G̃
(y(H̃ ∩ G̃)∗)

= ∨[CMG̃(z)], ∀z ∈ y(H̃ ∩ G̃)∗
≤ ∨[CMH̃◦G̃(z)], ∀z ∈ y(H̃∗ ∩ G̃∗)
≤ ∨[CMH̃◦G̃(z)], ∀z ∈ yH̃∗
= CMH̃◦G̃

H̃

(yH̃∗), ∀y ∈ G̃∗.

Hence, f( G̃
(H̃∩G̃)

) ⊆ H̃◦G̃
H̃

. Therefore, G̃
(H̃∩G̃)

' (H̃◦G̃)
H̃

.

Theorem 4.10. Let H̃, Ĩ, G̃ ∈ FMG(X) such that H̃ ⊆ Ĩ, and H̃ and Ĩ are
normal fuzzy submultigroups of G̃. Then ( G̃

H̃
)/( Ĩ

H̃
) ∼= G̃

Ĩ
.

Proof. If H̃, Ĩ ∈ FMG(X) and H̃ is a normal fuzzy submultigroup of Ĩ, then H̃∗
is a normal subgroup of Ĩ∗ and both H̃∗ and Ĩ∗ are normal subgroups of G̃∗. From
the principle of Third Isomorphism Theorem for groups, it follows that

(
G̃∗

H̃∗
)/(

Ĩ∗

H̃∗
) ∼= (

G̃∗

Ĩ∗
),

where f is given by

f(xH̃∗(
Ĩ∗

H̃∗
)) = xĨ∗, ∀x ∈ G̃∗.

256



Fuzzy comultisets and quotient fuzzy multigroups

Then

CM
f(( G̃

H̃
)/( Ĩ

H̃
))
(xĨ∗) = CM( G̃

H̃
)/( Ĩ

H̃
)
(xH̃∗(

Ĩ∗

H̃∗
))

= ∨[CM G̃
H̃

(yH̃∗)], ∀y ∈ G̃∗,yH̃∗ ∈ xH̃∗(
Ĩ∗

H̃∗
)

= ∨[∨(CMG̃(z)),∀z ∈ yH̃∗], ∀y ∈ G̃∗, yH̃∗ ∈ xH̃∗(
Ĩ∗

H̃∗
)

= ∨[CMG̃(z)], ∀z ∈ G̃∗,zH̃∗ ∈ xH̃∗(
Ĩ∗

H̃∗
)

= ∨[CMG̃(z)], ∀z ∈ xH̃∗(
Ĩ∗

H̃∗
)

= ∨[CMG̃(z)], ∀z ∈ G̃∗,f(z) ∈ xĨ∗
= ∨[CMG̃(z)], ∀z ∈ G̃∗,f(z) = z
= CMG̃

Ĩ

(xĨ∗),

∀x ∈ G̃∗, where the equalities hold since f is one-to-one. Hence, the result
follows.

5 Conclusions
In this paper, the ideas of fuzzy comultisets and quotient fuzzy multigroups

have been proposed in an attempt to further strengthen the theory of fuzzy multi-
groups. A number of some related results were duly discussed in details. The con-
nection between fuzzy comultisets of fuzzy multigroups and the cosets of groups
has been proven. Some characterizations of fuzzy comultisets were delineated
with verifications. In addition, quotient fuzzy multigroup was explicated and some
of its properties were explored. Finally, group theoretic isomorphism theorems
were substantiated in fuzzy multigroup setting. Nevertheless, some group analog
concepts could still be exploited in fuzzy multigroup context.

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Paul Augustine Ejegwa

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