Ratio Mathematica Volume 46, 2023

Anti Q-M-fuzzy normal subgroups

S.Palaniyandi *
R.Jahir Hussain†

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-
search scholars completed their research in field of fuzzy subgroup,
anti fuzzy subgroup, Q-fuzzy subgroup, anti Q-fuzzy subgroup, ho-
momorphism, anti homomorphism etc.
Keywords: Q-M-Fuzzy subgroup, Q-M- Fuzzy Normal Subgroups,
Anti Q-M- Fuzzy Normal Subgroups, group Q-M-Homomorphism
and group anti Q-M-Homomorphism.
AMS Subject Classification: 03E72, 03E75, 08A72 1

*PG & Research Department of Mathematics, Jamal Mohamed College, Affiliated to Bharathi-
dasan University, Tiruchirappalli, Tamilnadu, India. palanijmc85@gmail.com.

†PG & Research Department of Mathematics, Jamal Mohamed College, Affiliated to Bharathi-
dasan University, Tiruchirappalli, Tamilnadu, India. hssn jhr@yahoo.com.
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1071. ISSN: 1592-7415. eISSN: 2282-8214. ©S.Palaniyandi et
al. This paper is published under the CC-BY licence agreement.

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1 Introduction
According to Zadeh.L.A Rosenfeld [1971] was introduce fuzzy sets. It has

subsequently been employed in a variety of scientific domains, including engi-
neering, social science, medicine, and pure and applied mathematics. Rosenfeld
developed the concept of fuzzy subgroups Asaad [1991]. Biswas.R proposed anti-
fuzzy subgroups in Biswas [1990]. Solairaju.A and Nagarajan.R pioneered the
structure of Q-fuzzy groups Palaniappan and Muthuraj [2004]. Jacobson.N was
the first to use the words M-group and M-subgroup in Jacobson [1951]. In this
study, we present and discuss the concepts of group Q- M homomorphism, group
anti Q- M homomorphism, and anti Q- M-fuzzy normal subgroup of M-group.

2 Preliminaries
Definition 2.1. Let X 6= φ. Let X is fuzzy θ ⊆ θ : X ∈ [0,1].

Definition 2.2. Zadeh [1965] A fuzzy ⊆ θ ≤ N.
It is satisfying the axioms,

(i) θ (αβ) ≥ lower{θ (α) , θ (β)}

(ii) θ (α−1) = θ(α),∀α, β ∈ N.

Definition 2.3. Biswas [1990] Afuzzy ⊆ θ of a group G is said to be anti
fuzzy ⊆ G if it is satisfying the following conditions,

(i) θ (uv) ≤ lower{θ (u) , θ (v)}

(ii) θ (u−1) = θ(u),∀u, v ∈ G.

Definition 2.4. Biswas [1990] Let G: M and θ ⊆ G. Then θ is called M-fuzzy ⊆
G if ∀u ∈ G and m ∈ M, then θ(mx) ≤ θ(u)

Definition 2.5. Jacobson [1951] A Q-fuzzy set θ is Q−fuzzy ≤ G if ∀u,v ∈
G, and ρ ∈ Q

(i) θ (uv, ρ) ≥ lower{θ (u, ρ) , θ (v, ρ)}

(ii) θ (u−1, ρ) = θ (u, ρ)

Definition 2.6. Solairaju and Nagarajan [2009] Let fuzzy λ ⊆ X. For t�[0,1] ⊆
θ is denoted by [θt = {u ∈ U : θλ(u) ≥ t}]

Definition 2.7. Sithar Selvam et al. [2014] A Q-fuzzy set θ is called Q ≤ G if
∀u,v ∈ G, and ρ ∈ Q in anti Q fuzzy.

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(i) θ (uv,ρ) ≤ lower{θ (u,ρ) ,θ (v,ρ)}

(ii) θ (u−1,ρ) = θ (u,ρ)

Definition 2.8. Sithar Selvam et al. [2014] An antifuzzy normal Q ≤ G. Then
G ↗ θ of G if ∀x,y ∈ G and ρ ∈ Q, θ(uyx−1,ρ) = θ(v,ρ).

3 Anti Q-M- fuzzy normal subgroups and its level
subsets

Definition 3.1. Let θ be anti fuzzy Q−M−≤ M −groupG, then θ ≤ M(G) if
∀u,v ∈ G, ρ ∈ Q, and m ∈ M such that

θ(m(uvu−1),ρ) = θ(m(v),ρ) (or) θ(m(uv),ρ) = θ(m(vu),ρ).

Definition 3.2. Let θ be anti fuzzy Q−M ≤ M −groupG. For any t ∈ [0,1], the
subset θt is defined by θt = {u ∈ G,ρ ∈ Q,m ∈ Mθ(m(u),ρ) ≤ t} and it is the
subset of θ.

Theorem 3.1. If θ is a fuzzy Q−M− ⊆ of a M-group G, then θ is an anti fuzzy
Q−M−≤ M −groupG iff the level subset θt, t ∈ [0,1] is subgroup of M-group
G.

Proof. Let us assume that θ is an antifuzzy −Q−M−≤ M −groupG.
The level subset θt = {u ∈ G,ρ ∈ Q,m ∈ Mθ(m(u),ρ) ≤ t, t ∈ [0,1]}.
Let u,v ∈ θt, then θ(mx,ρ) ≤ t and θ(my,ρ) ≤ t
Now

θ(m(uy−1),ρ) ≤ upper{θ((mu),ρ),θ(m(v−1),ρ)}
= upper{θ(mu,ρ),θ(mv,ρ)}
≤ upper{t, t}

Thus

θ(m(uv−1),ρ) ≤ t

Hence xy−1 ∈ θt. Therefore θt ≤ M(G).
Conversely, Let θt be a subgroup of a M-group G.
Let u,v ∈ θt. Then θ(mu,ρ) ≤ t and θ(mv,ρ) ≤ t.

⇒ θ(m(uv−1),ρ) ≤ t , Because{uv−1 ∈ θt}
= upper{t, t}
= upper{θ(mu,ρ),θ(mv,ρ)}

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Therefore

θ(m(uv−1),ρ) ≤ upper{θ(mu,ρ),θ(mv,ρ)}

Hence θ is an anti fuzzy Q ≤ M −groupG.

Definition 3.3. Let θ be a anti fuzzy Q − M ≤ m − groupG. The set N(θ) is
defined by N(θ) = {α ∈ Gθ(m(αua−1),ρ) = θ(m(u),ρ)} ∀u ∈ G and ρ ∈ Q,
m ∈ M. and it is called an anti fuzzy Q-M-normalizer of θ.

Theorem 3.2. If θ is a fuzzy Q − M ≤ M − groupG. Then θ is an anti fuzzy
Q−M −fuzzy ≤ M −groupG iff the level subsets θt, t ∈ [0,1] ≤ M(G).

Proof. Let us assume that θ ≤ Q−M −antifuzzynormalsubgroupofaM(G)
and the level subsets θt, t ∈ [0,1] is a subgroup of a M-group G.
We take u ∈ G and α ∈ θt, then θ(ma,ρ) ≤ t
Now θ(m(αxa−1),ρ) = θ(ma,ρ) ≤ t.
Since θ is an anti fuzzy normal Q-M ≤ M(G), θ(m(uau−1),ρ) ≤ t
Therefore uαu−1 ∈ θt, hence θt ≤ M(G).

Theorem 3.3. If θ is an ≤ Q−M −antifuzzynormalsubgroupofaM(G)
Then

(i) N(θ) ≤ M(G).

(ii) θ is an normal anti fuzzy -Q-M ≤ iff N(θ) = G.

(iii) θ is an normal fuzzy anti−Q−M ≤ N(θ).

Proof. Let α,β ∈ N(θ). (i) Then θ(m(αua−1),ρ) = θ(mu,ρ)∀u ∈ G,ρ ∈
Q,m ∈ M and θ(m(βxβ−1),ρ) = θ(mu,ρ)∀u ∈ G,ρ ∈ Q,m ∈ M. Now

θ(m(αβu(αβ)−1),ρ) = θ(m(αβuβ−1α−1),ρ)

= θ(m(βuβ−1),ρ)

= θ(mu,ρ)

Then we have, θ(m(αβu(αβ)−1),ρ) = θ(mu,ρ)
⇒ αβ ∈ N(θ)
Therefore N(θ) ≤ M(G).
(ii) We know that

Nθ ⊆ G, (1)
θ is an normal fuzzy anti-Q−M ≤ G.
Let α ∈ G, then θ(m(αuα−1),ρ) = θ(mu,ρ) ∀ u ∈ G, ρ ∈ Q, m ∈ M.
Then

α ∈ N(θ) ⇒ G ⊆ N(θ) (2)

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From (1)&(2), we get
N(θ) = G
Conversely, assume that N(θ) = G
We have, θ(m(αxα−1),ρ) = θ(mx,ρ)∀α,x ∈ G,ρ ∈ Q,m ∈ M.
Therefore θ is an fuzzy normal anti Q−M ≤ M(G).
(iii) Let θ be an fuzzy normal anti Q−M ≤ M
We take α ∈ G, then we have
θ(m(αua−1),ρ) = θ(mu,ρ)∀u ∈ G,ρ ∈ Q,m ∈ M.
Therefore α ∈ N(θ) ⇒ G ⊆ N(θ).
Hence θ is an fuzzy normal anti Q−M ≤ N(θ)
Theorem 3.4. Let θ be an fuzzy normal anti Q−M ≤ M(G), then hθh−1 is also
an fuzzy normal anti Q−M ≤ M(G) ∀ h ∈ G, ρ ∈ Q, m ∈ M.
Proof. Given θ ≤ M(G) ≤ M(G)

(i)(hθh−1)(m(uv),ρ) = θ(m(h−1(uv)h),ρ)

= θ(m(h−1(uhh−1v)h),ρ)

= θ(m((h−1uh)(h−1vh)),ρ)

≤ upper{θ(m(h−1uh),ρ),θ(m(h−1vh),ρ)}
≤ upper{hθh−1(mu,ρ),hθh−1(mv,ρ)}

∀ u,v ∈ G, ρ ∈ Q and m ∈ M.

(ii)hθh−1(mu,ρ) = θ(m(h−1uh),ρ)

= θ(m(h−1uh)−1,ρ)

= θ(m(h−1u−1h),ρ)

= hθh−1(mu−1,ρ)

∀ u,v ∈ G, ρ ∈ Q , m ∈ M.
Therefore hθh−1 is an fuzzy anti Q−M ≤ M(G).
Theorem 3.5. Let θ fuzzy antiQ−M ≤ M(G), then hθh−1 fuzzy antiQ−M ≤
M(G), ∀h ∈ G, ρ ∈ Q, m ∈ M.
Proof. Given θ is an anti-Q-M-fuzzy normal subgroup of M-group G.
Then hθh−1 ≤ G.
Now

hθh−1(m(uvu−1),ρ) = θ(m(h−1(uvu−1)h),ρ)

= θ(m(uvu−1),ρ)

= θ(mv,ρ)

= θ(m(hvh−1),ρ)

= hθh−1(mv,ρ)

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Therefore hθh−1 is also an fuzzy normal anti Q−M ≤ M(G).

Theorem 3.6. The disjoint two fuzzy normal anti Q−M ≤ M(G) is also an anti
fuzzy anti Q−M ≤ M(G)G.

Proof. Let α and β be two anti-Q-M-fuzzy subgroups of a M-group G.
Then

(αβ)(m(uv−1),ρ) = lower{α(m(uv−1),ρ),β(m(uv−1),ρ)}
≤ lower{upper{α(c(u),ρ),α(m(v−1),ρ)}

,upper{β(c(u),ρ),β(m(v−1),ρ)}}
≤ lower{upper{α(cu,ρ),α(mv,ρ)},

upper{β(cu,ρ),β(mv,ρ)}}
≤ upper{lower{α(cu,ρ),α(mv,ρ)},

lower{β(cu,ρ),β(mv,ρ)}}

Therefore {(αβ)(c(uv−1),ρ)}≤ upper {(αβ)(cu,ρ),(αβ)(cv,ρ)}
Hence αβ is an fuzzy normal anti Q−M ≤ M(G).

Theorem 3.7. If C and D are an anti fuzzy normal Q-M ≤ M(G).Then A∩B is
anti fuzzy normal Q-M ≤ M(G).

Proof. For any x,y ∈ G, q ∈ Q, m ∈ M We have

(C ∩D)(m(xyx−1),q) = upper{C(m(xyx−1),q),D(m(xyx−1),q)}
= upper{C(my,q),D(my,q)}
= (C ∩D)(my,q)

Hence C∩ is an anti fuzzy normal Q-M ≤ M(G).

4 Group Q-M- homomorphism and group anti Q-
M- homomorphism

Definition 4.1. The function f : G×Q → H ×Q is homorphism group Q-M
(i) f : G → H is a homomorphism group and
(ii) f(m(uv),ρ) = (f(mv).f(mu),ρ) ∀ u,v ∈ G,ρ ∈ Q,m ∈ M. where G and
H are M-groups.

Definition 4.2. The function f : G×Q → H ×Q is anti homomorphism group
of Q-M if

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(i) f : G → H is homomorphism group

(ii) f(m(uv),ρ) = (f(mx).f(mv),ρ) ∀ u,v ∈ G,ρ ∈ Q,m ∈ M.

Theorem 4.1. If the function f : G × Q → H × Q is a group anti Q-M-
homomorphism

(i) If θ is an anti Q-M-fuzzy normal subgroup of H, then f−1(θ) is an fuzzy normal
anti Q−M ≤ M(G).

(ii) If f is an epimorphism and θ is an fuzzy normal anti Q−M ≤ M(G) then f(θ)
is an anti normal fuzzy Q-M ≤ H. Where G and H are M-groups.

Proof. (i) Given the function f : G × Q → H × Q is a group anti-Q-M-
homomorphism and θ is an anti normal fuzzy Q-M ≤ H.
For all u,v ∈ G,ρ ∈ Q,m ∈ M we have,

f−1(θ)(m(uvu−1),ρ) = θ(f(m(uvu−1)),ρ)

= θ(fm(u−1).f(mv).f(mu),ρ)

= θ(f(mv),ρ)

= f−1(mv,ρ)

Hence f−1(θ) is an fuzzy normal anti Q−M ≤ M(G).
(ii) Given θ is an fuzzy normal anti Q − M ≤ M(G). Then f(θ) is an anti

Q-M-fuzzy subgroup of H.
For any ℵ,β ∈ H, we have

f(θ)(m(αβℵ−1),ρ) = inf θ(mv,ρ) = infθ(m(uvu−1),ρ)
f(v) = αβα−1 = infθ(mv,ρ)

f(u) = a,f(v) = β = f(θ)(mb,ρ) (Since f is an epimorphism)

Therefore f(θ) is an fuzzy normal anti Q−M ≤ H.

Definition 4.3. Let A and B be two fuzzy anti Q−M ≤ M(G). The Product of A
and B is defined by AB(m(u),ρ) = infupper(A(mv,ρ),vz = u,B(mz,ρ))u ∈
G,ρ ∈ Q,m ∈ M.

Theorem 4.2. If A and B are fuzzy normal anti Q−M ≤ M(G), then AB is an
fuzzy normal anti Q−M ≤ G.

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Proof. Given A and B are two fuzzy normal anti Q−M ≤ M(G).

(i) AB(m(uv),ρ) = inf upper{A(m(u1y1),ρ),B(m(u2y2),ρ)}
where u = u1y1 and v = u2y2

≤ inf upper{upper{A(mu1,ρ),A(mv1,ρ)},
upper{B(mu2,ρ),B(mv2,ρ)}}

≤ upper{inf upper{A(mu1,ρ),A(mv1,ρ)},
inf upper{B(mu2,ρ),B(mv2,ρ)}}

i.e.,AB(m(uv),ρ) ≤ upper{AB(m(u1y1),ρ),AB(m(u2y2),ρ)}

(ii) AB(m(u−1),ρ) = infupper{B(m(z−1),ρ),A(m(v−1),ρ)}
where(vz)−1 = u−1

= inf upper{B(mz,ρ),A(mv,ρ)}
= inf upper{A(mv,ρ),B(mv,ρ)}
= AB(mv,ρ).

AB(mv−1,ρ) = AB(mv,ρ)

Hence AB is anti fuzzy normal Q-M ≤ M(G).

5 Conclusions
In this research article, we gave some results of anti Q-M-fuzzy normal

subgroup, Group Q-M homomorphism and Group anti Q-M homomorphism. This
article used to further research in fuzzy algebra.

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