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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue VI, June 2026
Construction of Q- Rung Orthopair (M, N) Uncertainty Level
Subgroup Structures
Rathinam Nagarajan
Department of Mathematics, J.J College of Engineering and Technology, Sowdambikka group of
institutions, Tiruchirappalli- 620009, Tamilnadu, India
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150600070
Received: 18 June 2026; Accepted: 23 June 2026; Published: 06 July 2026
ABSTRACT
The main objective of this article is to introduce the idea of q-rung orthopair (m, n)-fuzzy subgroups of a
finite group. We discuss the concept of q-rung orthopair (m, n) – fuzzy subgroups as a combination of
intuitionist fuzzy subgroup and Pythagorean fuzzy subgroup. We provide the definition of q-rung orthopair
(m, n) – fuzzy subgroup and examine various properties associated with it. Finally we analyze q-rung
orthopair (m, n)- fuzzy cosets, (m, n) – fuzzy normal subgroups and (m, n)- fuzzy level subgroups.
Keywords: intuitionist fuzzy set, Pythagorean fuzzy set, q-rung orthopair (m, n)- fuzzy subgroup, q-rung
orthopair (m,n)- fuzzy Coset, normal subgroup, homomorphism.
AMS Subject classification: 20N25, 03E72, 94605.
INTRODUCTION
To handle uncertainty in real-life problems Zadeh LA [1] proposed the concept of Fuzzy Sets (FSs). In 1971
expanding the notion of FSs, Rosenfeld A [2] defined fuzzy subgroup. In recent years many researchers
studied various properties of fuzzy subgroups, t-norm fuzzy subgroups, fuzzy level subgroup, fuzzy
subgroups and fuzzy homomorphism, anti-fuzzy subgroups, fuzzy normal subgroup, fuzzy coset and fuzzy
quotient subgroup etc [3-14]. In 1986, Atanassov KT [15] introduced Intuitionistic Fuzzy Set (IFS). In
1996, Intuitionistic fuzzy subgroup was first studied by Biswas R [16]. Zhan J et al. [17] introduced
intuitionistic fuzzy M-group. Furthermore, researchers developed intuitionistic fuzzy subgroup in many
ways [18-20]. In 2013, Yager RR [21] invented Pythagorean Fuzzy Sets (PFSs). In 2018, Naz .S et al. [22]
proposed a novel approach to decision-making problem using Pythagorean fuzzy set. In 2019, Akram M et
al. [23] applied
complex Pythagorean fuzzy set in decision-making problems. Ejegwa PA [24] gave an application of
Pythagorean fuzzy set-in career placements based on academic performance using max-min-max
composition. Some results related to it were given by Peng X [25] and Yang Y [26]. First time, Bhunia S et
al. [27] proposed “On the characterization of Pythagorean fuzzy subgroups”. In certain situations, the PFSs
unable to handle the problems like if membership degree η = 0.8 and nonmembership degree θ = 0.75 then 2
2 η θ+= ≥ 1.2025 1 it is clear that PFSs fail to define this type of sets but, 1 q q µ ν+ ≤ where q > 2 . Most
recently another amazing generalization of FSs is proposed by Yager RR [28], q-Rung Orthopair Fuzzy Sets
(q-ROFSs) which models uncertain and incomplete information better than both IFSs and PFSs with high
accuracy. q-ROFSs is more fruitful in many decision-making problems. This concept is perfectly designed
to represent imprecision and ambiguity in mathematical way and to produce a formalized tool to handle
fuzziness to real problems. Pinar A et al. [29] suggested a q-rung orthopair fuzzy multi-criteria group
decision making method for supplier selection based on a novel distance. Peng X et al. [30] published
Information measures for q-rung orthopair fuzzy sets. The main objective of this article is to introduce the
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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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idea of q-rung orthopair (m, n)-fuzzy subgroups of a finite group. We discuss the concept of q-rung
orthopair (m, n) – fuzzy subgroups as a combination of intuitionist fuzzy subgroup and Pythagorean fuzzy
subgroup. We provide the definition of q-rung orthopair (m, n) – fuzzy subgroup and examine various
properties associated with it. Finally we analyze q-rung orthopair (m, n)- fuzzy cosets, (m, n) – fuzzy
normal subgroups and (m, n)- fuzzy level subgroups.
Preliminaries :In this section, we recall a some definitions and ideas which are very helpful for further
proven.
Definition 2.1 Let A: X⟶[0,1] be a fuzzy subset of a group (X, 0).Then A is said to be a fuzzy subgroup of
(X,0) if the following conditions are exists.
1. A(w
1
, w
2
) ≥ min {A(w
1
), A(w
2
), ⩝ w
1
,w
2
∊ X
2. A(w
-1
) ≥ A(w) ⩝ w ∊ X
Definition 2.2 Let A = {(x, α(x), β(x)/x∊C} be a Ifs of a group (C, α), then A is said to be an IFSG of x if
the following conditions hold;
1. α (w
1
, w
2
) ≥ min {α(w
1
), α(w
2
)} and β(w
1
, w
2
) ≥ max {β(w
1
), β(w
2
)} ⩝ w
1
,w
2
∊ X
2. α (w
-1
) ≥ α(w) and β(w
-1
) ≥ β(w) ⩝ w∊ X
In 2017, yager defined q-rung orthopair fuzzy set as a speculation of IFS.
Definition 2.3: Let X be a crisp set. A q-rung orthopair fuzzy set A on X is defined by A = {(x,α(x), β(x)) /
x ∊ C} where α(x) ∊ [0,1] and β(x) ∊ [0,1] are the membership and non-membership degree of x ∊ X
respectively, which satisfy the condition D ≤ α
2
(x)+ β
2
(x) ≤ 1, (q ≥ 1) ⩝ W∊ X and its determinacy Π
A
= √1-
α
2
(x)+ β
2
(x) .
Definition 2.4: Basic operations on q-rung orthopair fuzzy set:Let A = { (x,α
1
(x), β
1
(x)/ x ∊ X} and B =
{ (x,α
2
(x), β
2
(x) /x ∊ X} be two q- ROFs of X-then the following holds.
1. A Ս B = {(x,α
1
(x) Ս α
2
(x), β
1
(x) Ո β
2
(x) /x ∊ X }
2. A Ո B = {(x,α
1
(x) Ո α
2
(x), β
1
(x) Ս β
2
(x) /x ∊ X}
3. A
C
=
{(x,α
2
(x), α
1
(x)/x ∊ X} and
4. A ⊆ B if a. α
1
(x), α
2
(x) and b. β
1
(x) = β
2
(x) ⩝ x ∊ X
5. A= B if a. α
1
(x) = α
2
(x) and b. β
1
(x) = β
2
(x) ⩝ x ∊ X
In this section let G be a finite group. This section elucidates the concept of a q-rung orthopair (m, n) –
fuzzy subgroup of G and delivers certain characteristics associated with it.
Definition 2.5 : Let A = { (w,γ
A
(w) )/ W∊G } be a q-rung orthopair (m, n)- fuzzy set on G, then A is a q-
rung orthopair (m, n) – fuzzy subgroup of G if, for w1, w2 ∊G;
(i) (δ
A
(m
1
w
1
,w
2
)))
q
≥ min { (δ
A
(w
1
)
q,
(δ
A
(w
1
)
q
} and
(A
A
(n
1
w
1
,w
2
)))
q
≥ max { (A
A
(w
1
)
q,
(A
A
(w
1
)
q
}
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(i) (δ
A
(n
1
w
1
)
q
= and (δ
A
(w
-1
)
q
and (A
A
(n
1
w
-1
))
q
= (A
A
(w
-1
))
q
,for m,n ∊ [0,1]
Note that if q=1 called intuitionistic fuzzy subgroup and q=2 Pythagorean fuzzy subgroup, q=3 fermatean
fuzzy subgroup of G.
Section :3 Standard Results on q-rung orthopair (m, n) Fuzzy subgroup
Theorem 3.1 Let A be a q- rung orthopair (m,n) – fuzzy subgroup of G, then
(δ
A
(w
k
))
q
≥ (δ
A
(w))
q
and (∆
A
(w
k
))
≥ (∆
A
(w))
q,
for
K∊N
Proof: By definition 2.5
(δ
A
(w
2
))
q
≥ min { (δ
A
(w))
q
,
(δ
A
(w))
q
} = (δ
A
(w))
q
and (∆
A
(w
2
))
q
≥ max {(∆
A
(w))
q
,
(∆
A
(w))
q
} = (∆
A
(w))
q
The proof follows by mathematical induction.
Corollary 3.2 In definition 2.5 since G is a finite group, condition 1 gives condition 2
Proof Let w ∊ G with order of W.
O( W ) = S, then w
-1
= w
n-1
, (δ
A
(m
-1
))
q
≥ (δ
A
(mw
n-1
)
q
≥ (δ
A
(w)
q
)
For w = w
-1
if follows
that (δ
A
(mw))
q
= (δ
A
(w
-1
)
q
Similarly , for non-membership degree (A
A
(mw))
q
= (A
A
(w
-1
))
q
Theorem 3.3 Let A be a q-rung orthopair (m, n)- fuzzy subgroup of G, ‘e’ be the identify element in
G.Then (δ
A
(me))
q
≥ (δ
A
(w)
q
and
(A
A
(ne))
q
≤ (A
A
(w))
q,
for w ∊ G
Proof: Let w ∊ G, (δ
A
(me))
q
= (δ
A
(ww
-1
)
q
≥ min { (δ
A
(w))
q
, (δ
A
(w
-1
)
q
} ≥ (δ
A
(w))
q
Analogously, it can be shown that (A
A
(ne))
q
≤ (A
A
(w))
q
Theorem 3.4 Let A be a q-rung orthopair (m,n) – fuzzy subgroup of G if and only if
(δ
A
(m(w
1
w
2
-1
)
q
≥ min { (δ
A
(w
1
))
q
, (δ
A
(w
2
)
q
} and (∆
A
(n(w
1
w
2
))
q
} ≤ max { (∆
A
(w
1
))
q
, (∆
A
(w
2
))
q
},for w
1
w
2
∊
G and min ∊ [0,1]
Proof Let A be a q-rung orthopair (m, n)-fuzzy set on a group G.
Suppose A is a q-rung orthopair (m, n)- fuzzy subgroup of G, then for w
1
w
2
∊ G,
(δ
A
(m(w
1
w
2
-1
)))
q
≥ min { (δ
A
(w
1
))
q
,
(δ
A
(w
1
-1
))
q
} = min { (δ
A
(w
1
))
q
,
(δ
A
(w
1
))
q
} and
(∆
A
(n(w
1
, w
2
-1
))
≤ max {(∆
A
(w
1
))
q
, (∆
A
(w
2
-1
))
q
} = max {(∆
A
(w
1
))
q
, (∆
A
(w
2
))
q
}
Conversly,
Let (δ
A
(m(w
1
w
2
-1
)))
q
≥ min { (δ
A
(w
1
)) }and (∆
A
(n(w
1
, w
2
-1
))
q
≤ max {(∆
A
(w
1
))
q
},
for all w
1
w
2
∊ G .Then
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(δ
A
(m(w
1
w
2
)))
q
= (δ
A
(m(w
1
(w
2
)) )
q
≥ min { (δ
A
(w
1
))
q
,
(δ
A
(w
2
-1
)
q
}
= min { (δ
A
(w
1
))
q
,
(δ
A
(w
2
)
q
}
Similarly it can be proved that (∆
A
(n(w
1
, w
2
)
))
q
≤ max {(∆
A
(w
1
))
q
, (∆
A
(w
2
))
q
}
Now consider (δ
A
(m(w
1
-1
))
q,
(δ
A
(m(w
1
-1
)))
q
= (δ
A
(m(cw
1
-1
)))
q
≥ min {(δ
A
(e))
q
, (δ
A
(w
1
))
q
}
= (δ
A
(w
1
))
q
........... (1)
Which will give
(δ
A
(m(w
1
))
q
= (δ
A
(m(w
1
-1
))
q
≥ (δ
A
(w
1
-1
))
q
............... (2)
From (1) and (2), (δ
A
(w
1
-1
))
q
=
(δ
A
(m(w
1
))
q .
Similarly, (∆
A
(n(w
1
-1
))
q
= (∆
A
(w
1
))
q
Hence ‘A’, is a q-rung ortho pair (m, n)- fuzzy subgroup of G.
Theorem 3.5 Let A and B be two q-rung orthopair (m, n) – fuzzy subgroup of G then
A Ո B is a q-rung orthopair (m, n) - fuzzy subgroup of G.
Proof : For w
1,
w
2
∊ G
(δ
A Ո B
(m(w
1
w
2
-1
)))
q
= min { (δ
A
(m(w
1
w
2
-1
)))
q
, (δ
B
(m(w
1
w
2
-1
)))
q
}
≥ min{ min {(δ
A
(w
1
))
q
,( δ
A
(w
1
-1
))
q
}, min {(δ
B
(w
2
))
q
,( δ
B
(w
2
))
q
}}
= min{ min {(δ
A
(w
1
))
q
,( δ
B
(w
1
))
q
}, min {(δ
A
(w
2
))
q
,( δ
B
(w
2
))
q
}}}
= min{ δ
A Ո B
(w
1
), δ
A
⨻
B
(w
2
)}
Similarly it can be shown that
(∆
A Ո B
(n(w
1
, w
2
-1)
))
q
≤ max {(∆
A Ո B
(w
1
))
q
, (∆
A Ո B
(w
2
))
q
}
Hence
A Ո B
is a q-rung orthopair(m ,n) fuzzy subgroup of G.
Theorem 3.6 Let “A’ be a q-rung orthopair (m, n) – fuzzy subgroup of G, e be the element in G then
(δ
A
(m(xw)))
q
= (δ
A
(w))
q
and (∆
A
(n(xw)))
q
= (∆
A
(w))
q
for all w
1
w
2
∊ G.If and only if (δ
A
(x))
q
= (δ
A
(e))
q
and
(∆
A
(x))
q
= (∆
A
(e))
q
Proof Suppose (δ
A
(m(xw)))
q
= (δ
A
(w))
q
and (∆
A
(n(xw)))
q
= (∆
A
(w))
q
for all w
1
w
2
∊ G
Specifically, w = e, it follows that (δ
A
(x))
q
= (δ
A
(e))
q,
(∆
A
(x))
q
= (∆
A
(e))
q
Suppose
(δ
A
(x))
q
= (δ
A
(e))
q
and
(∆
A
(x))
q
= (∆
A
(e))
q
Since (δ
A
(mw))
q
≤ (δ
A
(e))
q
for all w
1
w
2
∊ G , (δ
A
(m(xw)))
q
≥ min{ (δ
A
(x))
q
,
(δ
A
(w))
q
}
and
= min{ (δ
A
(e))
q
,
(δ
A
(w))
q
} = (δ
A
(w))
q
for all w
1
w
2
∊ G
Also, (δ
A
(w))
q
=
( α
A
(x
-1
w))
q
} ≥ min {(δ
A
(x))
q
,
(δ
A
(xw))
q
} = (δ
A
(w))
q
for all w
1
w
2
∊ G
Hence, (δ
A
(mw))
q
= (δ
A
(xw))
q
for all w
∊ G.Similar, (∆
A
(xw))
q
= (∆
A
(mw))
q
for all w
∊ G
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Theorem 3.7 Let “A’ be a q-rung orthopair (m, n) – fuzzy subgroup of G, e be the element in G then H= {
w
∊ G/ (δ
A
(mw))
q
= (δ
A
(e))
q
, (∆
A
(mw))
q
= (∆
A
(e))
q
} is a subgroup of G.
Proof: By definition of H, it follows that e ∊ H. Hence, H is a non – empty subset of G. Let w1, w2 ∊ H.
Then (δ
A
(mw
1
))
q
= (δ
A
(e))
q
= (∆
A
(mw
2
))
q
and (∆
A
(mw
1
))
q
= (∆
A
(e))
q
= (∆
A
(mw
2
))
q
. we have
(δ
A
(m(w
1
w
2
-1
)))
q
≥ min { (δ
A
(w
1
)
q
, (δ
A
(w
2
-1
))
q
} = min { (δ
A
(w
1
)
q
, (δ
A
(w
2
))
q
}
= min { (δ
A
(e)
q
)
, (δ
A
(e))
q
} = (δ
A
(e))
q.
By theorem 3.3
(δ
A
(e))
q
≥ (δ
A
(m(w
1
w
2
-1
))
q
therefore (δ
A
(m(w
1
w
2
-1
))
q
=(δ
A
(e))
q
Similarly it can be shown that (∆
A
(m(w
1
w
2
-1
))
q
=(∆
A
(e))
q
thus w
1
w
-1
2
∊ H.
Hence it is a subgroup of G.
Theorem 3.8 Let “A’ be a q-rung orthopair (m, n) – fuzzy subgroup of G, then there exists an element x ∊
G such that (δ
A
(x))
q
≤
(δ
A
(mw))
q
and (∆
A
(x))
q
≤
(∆
A
(mw))
q
for all
w
1
w
2
∊ G.
Proof : Let k
1
denote the set of all elements in G with the lowest membership degree, and K
2
denote the set
of all elements in G with the highest non-membership degree, if
k
1
= { w
1
∊ G/ (δ
A
(mw
1
))
q
≤ (δ
A
(w))
q
⩝ w ∊ G } and
K
2
= { w
11
∊ G/ (δ
A
(mw
11
))
q
≤ (∆
A
(w))
q
⩝ w ∊ G }
To show K
1 Ո
K
2
≠ ∅. Let w
1
∊ K
1 and
w
11
∊ K
2
Clearly w
1
= w
11
∊ p for some P ∊ G and (δ
A
(mw
-1
))
q
≥ min {(δ
A
(w
11
))
q
,
(δ
A
(p))
q
}
Since, (δ
A
(w
11
))
q
is the lowest membership
degree
either (δ
A
(w
11
))
q
=
(δ
A
(w
1
))
q
or
(δ
A
(mp))
q
= (δ
A
(mw
1
))
q
if (δ
A
(mw
11
))
q
= (δ
A
(mw
1
))
q
, it follows that
w
11
∊ K
1
therefore K
1 Ո
K
2
≠ ∅ and x = w
11.
if (δ
A
(mp))
q
= (δ
A
(mw
11
))
q
, consider w
11 =
w
1
* p
-1
then
(∆
A
(m
w
11
))
q
≤ max {(∆
A
(w
11
))
q
, (∆
A
(p
-1
))
q
}.
Since (∆
A
(m w
11
))
q
is the highest non-membership degree ,then
(∆
A
(nw
1
))
q =
(∆
A
(nw
11
))
q
or
(∆
A
(np
-1
))
q =
(∆
A
(nw
11
))
q
if
(∆
A
(nw
1
))
q =
(∆
A
(nw
11
))
q
if follows that w
11
∊
K
2
, K
1 Ո
K
2
≠∅ and x = w
1
if (∆
A
(np
-1
))
q
= (∆
A
(nw
11
))
q
, then it follows
that (δ
A
(mp))
q
= (δ
A
(mw
1
))
q
and (∆
A
(mp
-1
))
q
= (∆
A
(mw
11
))
q
,
which means p ∊ K
1 Ո
K
2
and x = ∅.
Theorem 3.9 : Let G be a cyclic group and let A be a q-rung orthopair (m, n) – fuzzy subgroup of G. Then
the generators of G posses and membership and non-membership degree in K.
Proof: let W
1,
W
2
be
two generators of G. Since W1 is a generator, W
2
= W
1
r
for some
r ∊ N .(δ
A
(mw
2
))
q
= (δ
A
(mw
1
r
))
q
≥ (δ
A
(w
1
))
q
and (∆
k
(mw
2
))
q
= (∆
k
(mw
1
r
))
q
≤ ∆
A
(w
1
)
q
⤏(3)
Since w2 is a generator, w
1
= w
2
for some S ∊N.
Page 1004
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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue VI, June 2026
(δ
A
(mw
1
))
q
= (δ
A
(mw
2
s
))
q
≥ (δ
A
(w
2
))
q
and (∆
A
(nw
1
))
q
= (∆
A
(nw
2
s
))
q
≤ ∆
A
(w
2
)
q
⤏(4)
From (3) and (4) (δ
A
(mw
1
))
q
= (δ
A
(w
2
s
))
q
and (∆
A
(nw
1
))
q
= (∆
A
(w
1
))
q
Some Orders of Q-Rung Orthopair (M, N) – Fuzzy Subgroups
Definition 4.1: Let ‘A’ be a q-rung orthopair fuzzy subgroup of group G. For W ∊ G, the least positive
integer ‘r’ such that (δ
A
(mw
r
))
q
= (δ
A
(e))
q
and (∆
A
(nw
r
))
q
= (∆
A
(e
))
q
is called the
order of W in A, denoted as
order of q-rung orthopair(m, n)-fuzzy subgroup.
Remark 4.2 The order of an element in a q-rung ortho pair (m, n) – fuzzy subgroup A of G is always less
than or equal to its order in G. Also the order of an element and its inverse in ‘A’ are equal.
Theorem 4.3: Let A be a q-rung orthopair (m, n) – fuzzy subgroup of G, and ‘e’ be the identity element in
G. Let w
1
∊ G, and let ’u’ be a (t) be in satisfying. (δ
A
(mw
1
u
))
q
= (δ
A
(e))
q
and (∆
A
(nw
1
u
))
q
= (∆
A
(e
))
q
then
order of q-rung orthopair (m, n) – fuzzy subgroup divides ‘u’.
Conclusion: This objective of this article introduce the concept of q-rung orthopair (m, n)- fuzzy subgroups
of finite groups and examines their fundamental properties. Additionally, it develop and analyzes key
concepts such q-rung orthopair (m, n)- fuzzy subgroup, there by providing deeper insights in to the structure
of q-rung orthopair (m, n)- fuzzy subgroups.
Future work : The research may explore concepts such as q-rung orthopair (m, n)- fuzzy homomorphism
and q-rung orthopair (m, n)- fuzzy isomorphisms..
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue VI, June 2026
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