Ratio Mathematica Volume 46, 2023

Anti-homomorphism in Q-fuzzy subgroups
and normal subgroups

R.Jahir Hussain*
S.Palaniyandi†

Abstract

The fuzzy set has been applied in wide area by many researchers.
We define the concept of anti-homomorphism in Q-fuzzy subgroups
and Q-fuzzy normal subgroups and establish some result in this re-
search article and develop some theory of anti- homomorphism in
Q-fuzzy subgroups, normal subgroups and also extend results on Q-
fuzzy abelian subgroup and Q- fuzzy normal subgroup. Many re-
searchers have explored the fuzzy set extensively. We propose the
notion of anti-homomorphism in Q is fuzzy subgroups and normal
subgroups. It is establish some findings in this study article and build
the theory of anti-homomorphism in Q-fuzzy subgroups, normal sub-
groups. It is also extend results on Q-fuzzy abelian subgroup and
Q-fuzzy normal subgroup.
Keywords: Fuzzy, subgroup, Q-fuzzy, fuzzy abelian, fuzzy normal
subgroup, anti-homomorphism,.
AMS Subject Classification: 03E72, 03E75, 08A72 1

*PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Af-
filiated to Bharathidasan University, Tiruchirappalli, Tamilnadu, India.; hssn jhr@yahoo.com.

†PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Af-
filiated to Bharathidasan University, Tiruchirappalli, Tamilnadu, India.; palanijmc85@gmail.com.
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1073. ISSN: 1592-7415. eISSN: 2282-8214. ©R.Jahir Hussain
et al. This paper is published under the CC-BY licence agreement.

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R.Jahir Hussain and S.Palaniyandi

1 Introduction
Zadeh L.A. Zadeh [1965] introduced the fuzzy set concept. Numerous schol-

ars have used the fuzzy set in several different contexts. Fuzzy subgroups are
first discussed by Rosenfeld Rosenfeld [1971]. Biswas.R Biswas [1990] was in-
troduced, the anti-fuzzy subgroups. The novel structure of Q-fuzzy subgroups
was introduced by Solairaju.A and Nagarajan.R Solairaju and Nagarajan [2009].
Fuzzy subgroups and fuzzy homomorphisms were defined by Choudhury, F.P.,
Chakraborty, A. B, and Khare Choudhury et al. [1988] Sheik Anti-homomorphism
in fuzzy subgroups was defined by Abdullah A. and Jeyaraman K. Sheik Abdullah
and Jeyaraman [2010]. In this study, we demonstrate various results and define the
notion of anti-homomorphism in Q-fuzzy subgroups and fuzzy normal subgroups.

2 Preliminaries
Definition 2.1. Zadeh [1965] A function of fuzzy subset δ 6= S is δ : S → [0,1].

Definition 2.2. Rosenfeld [1971] A fuzzy subset δ of a group J = J (fuzzy sub-
group) if it is satisfying the following conditions,

(i) δ (%) ≥ min{δ (%) , δ (γ)}

(ii) δ (%−1) = δ(%), ∀ %, γ ∈ J.

Definition 2.3. Solairaju and Nagarajan [2009] A Q-fuzzy set δ = J if ∀ %,
gammaf ∈ J, and κ ∈ Q

(i) δ (%γ, κ) ≥ min{δ (%, κ) , δ (γ, κ)}

(ii) δ (%−1, κ) = δ (%, κ)

Definition 2.4. Zadeh [1965] ν ⊆ S (fuzzy subset of a set) . For β ∈ [0,1], the
level subset of δ is defined by

δβ = {e ∈ S : δν(%) ≥ β

Definition 2.5. Solairaju and Nagarajan [2009] ν ⊆ S. For β ∈ [0,1], the set
δβ = {e ∈ S, κ ∈ Q : δν(%, κ) ≥ β} is called a Q ⊆ δ.

Definition 2.6. Palaniappan and Muthuraj [2004] Consider δ < J. The fuzzy
subgroup δ is said to be fuzzy normal subgroup if δ (%γ) = δ (fe), ∀ %, γ ∈ J.

Definition 2.7. Palaniappan and Muthuraj [2004] A fuzzy subgroup δ of a group
J is a Q-fuzzy normal subgroup if δ (%γ, κ) = δ (γ%, κ), ∀ varrho γ ∈ J, and
κ ∈ Q.

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Anti-homomorphism in Q-fuzzy subgroups and normal subgroups

Definition 2.8. Choudhury et al. [1988] Let (J1, •) and J2, •) be the function
g : J1 → J2 is called a group homomorphism if g (%γ) = g (%) .g(γ), ∀%, γ ∈ J1.
Definition 2.9. Sheik Abdullah and Jeyaraman [2010] Let (J1, •) and J2, •)
be the function g : J1 → J2 is called a group anti homomorphism if g (%γ) =
g (γ) .g (%), ∀ %, γ ∈ J1.
Definition 2.10. Sheik Abdullah and Jeyaraman [2010] Let g : J1 → J2 is called
anti automorphism if g (%γ) = g (f) .g(%) ∀ %, γ ∈ J1.
Definition 2.11. Sheik Abdullah and Jeyaraman [2010] The function δ is a fuzzy
characteristic subgroup of a group J if δ (h(%)) = δ(%).

3 Some results On Q -fuzzy subgroups in anti- ho-
momorphism

Theorem 3.1. Let g : J → J∗ be an anti-homomorphism, if δ∗ is a Q<J∗. Then
g−1(δ∗) is a Q < J.

Proof. Let e, γ ∈ J. Then

g−1 (δ∗) (%γ, κ) = δ∗(h(%γ, κ))

= δ∗{g (γ, κ) .g (%, κ)}
≥ min{δ∗ (h(γ, κ)) , δ∗(h(%, κ))}
= min{(h−1δ

∗
)(γ, κ), (h

−1
δ
∗
)(%, κ)} (1)

and

g−1 (δ∗)
(
%−1, κ

)
= δ∗(h

(
%−1, κ

)
)

= δ∗(h(%, κ))

= g−1 (δ∗) (%, κ) (2)

From (1) and (2), g−1(δ∗) is a Q < J.

Theorem 3.2. If δ is a Q <J and g : S → S∗ is an anti-homomorphism, then
g−1(δ) is a Q-fuzzy normal subgroup of S∗.

Proof. For every %, γ ∈ S. We get,

g−1 (δ) (%, κ) = δ(h(%, κ)

= δ{g (γ, κ) , g (%, κ)}
= δ(h(γ%, κ)

= g−1 (δ) (γ%, κ)

Hence g−1(δ) is a Q <J.

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R.Jahir Hussain and S.Palaniyandi

Theorem 3.3. A fuzzy characteristic subgroup of a Q < Q. It is a fuzzy normal
subgroup.

Proof. Given g is an anti automorphism of S. For all %, γ ∈ S, and κ ∈ Q. Then

g (%γ) = g (γ) .g(%), for every e, γ ∈ S

Now,

δ (%γ, κ) = δ(h(%γ, κ)

= δ{g (f, κ) ,g (%, κ)}

Since δ is a characteristics Q <S.
Then δ (%γ, κ) = δ{g (γ, κ) ,g (%, κ)}
Since g is anti automorphism of S,

δ (%γ, κ) = δ(h(fe, κ))

= δ(γ%, κ), for all %, γ ∈ S, and κ ∈ Q

δ is a characteristics Q <S.
Hence δ is a Q <S.

Definition 3.1. The Q-fuzzy subgroup δ of a group S is called a Q -fuzzy abelian
subgroup of S if H = {% ∈ S : δ (%, κ) = δ (i, κ)}, ∀ %, γ ∈ S, and κ ∈ Q.

Theorem 3.4. The commutative property satisfies all anti-homomorphism pre-
images of a Q-fuzzy commutative subgroup.

Proof. If δ is a Q <S.
To prove δ is a Q=[S, S]
Let us consider ν is a Q = [S∗, S∗] (Q fuzzy commutative subgroup of S∗).
Since δ is a Q = [S∗, S∗] and ν is Q = [S∗, S∗]
Then V = {γ ∈ S∗, κ ∈ Q : ν (γ, κ) = ν (i∗,κ)} is a Q = [S∗, S∗], where i∗ is
the identify element of S∗.
Let T = {% ∈ S, κ ∈ Q : δ (%, κ) = δ (i, κ)} where i is the identify element
of S.
Take %, γ ∈ T this implies %γ ∈ T ⊆ S. Then

δ (%γ, κ) = δ(i, κ)

ν (h(%γ, κ)) = ν(h(i,κ))

= ν (i∗,κ)

ν{g (γ, κ) .g (%, κ)} = ν (i∗,κ)

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Anti-homomorphism in Q-fuzzy subgroups and normal subgroups

Since g (γ, κ) .g (%, κ) ∈ V and V is abelian,

g (f, κ) .g (%, κ) = g (%, κ) .g (γ, κ)

ν(h(γ, κ) .g (%, κ)) = ν(h(%, κ) .g (γ, κ))

ν (γ (γ%, κ)) = ν(γ (%γ, κ)), since g is anti−homomorphism.
δ (%γ, κ) = δ (γ%, κ)

δ (i, κ) = δ (γ%, κ)

i.e (γ%, κ) = δ (i, κ) , this implies fe ∈ T and κ ∈ Q.
For all %, γ ∈ T , %γ ∈ T and γ% ∈ T .
This implies %γ = γ%
T satisfies commutative. Therefore δ satisfies commutative property.

Theorem 3.5. Anti-homomorphism image of a Q -fuzzy commutative subgroup is
also satisfies commutative level.

Proof. Let ν be a Q -fuzzy subgroup of S∗.
To prove: ν is a Q -fuzzy commutative subgroup of S∗.
Let g be an anti-homomorphism from S to S∗.
Since δ is a Q=[S, S].
Then T = {% inS, κ ∈ Q : δ (%, κ) = δ (i, κ)} is an commutative Q -fuzzy
subgroup of S∗ where i is the identity element of S.
Let ν be the Q - fuzzy subgroup of S∗.
Let V = {x ∈ S∗, κ ∈ Q : ν (%, κ) = δ (i∗, κ)} where i∗is the identity element
S∗.
Let e, γ in V ⊆ S∗

ν (%γ, κ) = ν(i∗, κ)

Sup δ (r, κ) = Sup δ (r, κ)

r ∈ g−1(%γ) , r ∈ g−1(i∗)
δ (%γ, κ) = δ(i, κ)

Then %γ ∈ T and T is an commutative Q - fuzzy subgroup.

(%γ,κ) = (γ%, κ)

δ (%γ,κ) = δ(γ%, κ)

Sup δ (r, κ) = Sup δ (r, κ)

r ∈ g−1(%γ) , r ∈ g−1(γ%)
ν (%γ,κ) = ν(γ%, κ)

ν (i∗,κ) = ν(γ%, κ)

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R.Jahir Hussain and S.Palaniyandi

That is ν (%γ,κ) = ν (i∗,κ), this implies γ% ∈ V and κ ∈ Q.
For all %, γ ∈ V , %γ ∈ V
This implies γ% ∈ V and %γ = γ%.
Then V = [S∗, S∗] (commutative subgroup of S∗).
Therefore ν = [S∗, S∗]. Where Q fuzzy commutative subgroup of S∗.

4 Conclusions
Many results can be found from the research article. But, in this paper we

found few concepts of anti-homomorphism in Q-fuzzy subgroups. Further this
paper used to developing the concept of Q-fuzzy abelian subgroup. There are so
many concepts can be availed by future research work.

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N. Palaniappan and R. Muthuraj. Anti fuzzy group and lower level subgroups.
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A. Rosenfeld. fuzzy groups. J. math. Anal.Appl., 35(3):512–517, 1971.

A. Sheik Abdullah and K. Jeyaraman. Anti- homomorphism in fuzzy subgroups.
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