Int. J. Anal. Appl. (2022), 20:55

Bipolar Fuzzy Magnified Translations in Groups

U. Venkata Kalyani1,∗, T. Eswarlal1, K. V. Narasimha Rao2, A. Iampan3

1Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education

Foundation, Vaddeswaram, AP, India
2Department of Mechanical Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram,

Guntur, AP, India
3Department of Mathematics, School of Science, University of Phayao, Maeka, Mueang, Phayao

56000, Thailand

∗Corresponding author: u.v.kalyani@gmail.com

Abstract. In this paper, we define a bipolar fuzzy magnified translation (BFMT) of a bipolar fuzzy

subgroup (BFSG) of a group. Based on this concept we have also developed some important results

and theorems on bipolar fuzzy groups.

1. Introduction

The notion of a fuzzy set (FS) was introduced in 1965 by Zadeh [16]. The FS theory has various

expansions, such as intuitionistic fuzzy sets (IFS), interval-valued fuzzy sets (IVFS), vague sets (VS),

and so on. With, the traditional FS representation it is not easy to explicitly express the difference

of the irrelevant elements from the contrary elements. Based on these observations, in 2000, Lee [1]

introduced a bipolar valued fuzzy set (BVFS), an extension of FSs whose range of the membership

degree (MSD) is enlarged from [0,1] to [−1,1]. Kalyani and Eswarlal [6–10] have introduced and
studied the bipolar vague cosets, normal groups, bipolar fuzzy sublattices,ideals and also gave the

application of TOPSIS and ELECTRE1 method on bipolar vague sets. The idea of fuzzy magnified

translation (FMT) has been coined by Majumder and Sardar [15] in 2008. Jun [14] explored the

bipolar fuzzy translations (BFT) in BCK/BCI-algebras in 2009. Kumar [13] popularized bipolar valued

fuzzy translations (BVFT) in semigroups in 2012. The notion of an intuitionistic fuzzy magnified

Received: Aug. 28, 2022.

2010 Mathematics Subject Classification. 06D50, 06D72, 03E72.

Key words and phrases. bipolar fuzzy set; bipolar fuzzy sublattice; bipolar fuzzy ideal; homomorphism.

https://doi.org/10.28924/2291-8639-20-2022-55
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-55


2 Int. J. Anal. Appl. (2022), 20:55

translation (IMFT) in groups has been discussed by Sarma [12] in 2012. The notion of translation

(T), multiplication (M), and extension (E) applied in distinct aspects on different structures in algebra.

In 2019, Iampan [4] studied translation (T) and density (D) of a bipolar valued fuzzy set (BFS) in

UP-algebras. Anggraenil [5] has given BFT(bipolar fuzzy translation), BFE(bipolar fuzzy extension),

and BFM(bipolar fuzzy multiplication) on bipolar anti-fuzzy ideals of K-algebras in 2019. In 2021,

Alshehri [2] studied fuzzy translation (FT) and multiplication (FM) in BRK algebras, Khamrot [3]

studied on a right weakly regular semigroup (RWRSG) of generalized a bipolar fuzzy subsemigroup

(BFSSG).

Here in this paper, we introduce a Bipolar fuzzy magnified translation in groups and obtained some

interesting results.

Throughout the paper FS stands for a fuzzy set, FG stands for a fuzzy subgroup, BFS stands for

a bipolar fuzzy subset, BFSG stands for a bipolar fuzzy subgroup, BFNSG stands for a bipolar fuzzy

normal subgroup, BFMT stands for a bipolar fuzzy magnified translation, G is always a group and D
is universe of discourse.

2. Preliminaries

Here, we will review a few standard definitions that are relevant to this work.

Definition 2.1. [16] A mapping δ :Z→ [0,1] is represented to as a fuzzy subset (FS) of a nonempty
set Z.

Definition 2.2. [1] A BFS B in D is an object having the form B= {< T ,BN(T ),BP(T ) >: T ∈D},
where BP : D → [0,1] and BN : D → [−1,0]. The positive membership degree (+ve MSD) BP(T )
denotes the satisfaction degree of an element T to the property corresponding to B and the negative
membership degree (-ve MSD) BN(T ) denotes the satisfaction degree of T to some implicit counter
property of B. For the sake of simplicity, we shall use the symbol B =< BN,BP > for the BFS
B= {< T ,BN(T ),BP(T ) >: T ∈D}.

Definition 2.3. [11] A BFS B of G is said to be a BFSG of G if the following conditions are satisfied:
(i) BP(ξη)≥min{BP(ξ),BP(η)},
(ii) BP(ξ−1)≥BP(ξ),
(iii) BN(ξη)≤max{BN(ξ),BN(η)}, and
(iv) BN(ξ−1)≤BN(ξ) for all ξ,η in G.

Definition 2.4. [11] A BFSG B of G is said to be a BFNSG of G if the following conditions are
satisfied:

(i) BP(ξη)=BP(ηξ) and
(ii) BN(ξη)=BN(ηξ) for all ξ,η in G.

Definition 2.5. [11] Let η be a mapping from a group K to a group K′ and let A=(K;AN,AP) BFS in
K and B=(K′;BN,BP) BFS in η(g)=K′ defined by BP(h)= sup{AP(g)} and BN(h)= inf{AN(g)},



Int. J. Anal. Appl. (2022), 20:55 3

where g ∈ η−1(h) for all g ∈K and h ∈K′. A is called the preimage of B under η and is denoted by
η−1(B) and is defined by for g ∈K, (η−1(BP(g)))=BP(η(g)) and (η−1(BN(g)))=BN(η(g)).

Remark 2.1. [4] For any BFS B=< BN,BP > in D, we denote 5=−1− inf{BN(T ) : T ∈D} and
4 = 1− sup{BP(T ) : T ∈ D}. Let B =< BN,BP > be a BFS in D and (θ,ϑ) ∈ [5,0]× [0,4].
By a bipolar fuzzy (θ,ϑ)-translation of B=< BN,BP >, we mean a BFS BT

(θ,ϑ)
=< BN

(θ,T)
,BP
(ϑ,T)

>,

where BN
(θ,T)

: D → [−1,0] defined by BN
(θ,T)
(T ) = BN(T )+ θ and BP

(ϑ,T)
: D → [0,1] defined by

BP
(ϑ,T)
(T )=BP(T )+ϑ for all T ∈D.

3. Bipolar fuzzy magnified translations in groups

Definition 3.1. Let B =< BN,BP > be a BFS in D and (α,β) ∈ [0,1], (θ,ϑ) ∈ [5,0]× [0,4]. By
a BFMT of B =< BN,BP >, we mean a BFS M = {< r,BN

(α,θ)
(r),BP

(β,ϑ)
(r) >: r ∈ D} or simply as

M = {< r,BNM(r),B
P
M(r) >: r ∈ D}, where B

N
M = B

N
(α,θ)

: D → [−1,0] and BPM(r) = B
P
(β,ϑ)

: D →
[0,1] defined by BNM(r)=B

N
(α,θ)
(r)= αBN(r)+θ and BPM(r)=B

P
(β,ϑ)
(r)= βBP(r)+ϑ for all r ∈D.

Example 3.1. Let D= {1,ω,ω2} and let B= {< 1,−0.2,0.3 >,< ω,−0.3,0.4 >,< ω2,−0.1,0.5 >
}. Then θ ∈ [−0.9,0] and ϑ ∈ [0,0.5]. Let α = 0.1,β = 0.2,θ = −0.8,ϑ = 0.2. Hence, a BFMT
M = {< 1,−0.8,0.26 >, < ω,−0.83,0.36 >,< ω2,−0.81,0.3 >}.

Theorem 3.1. Let M be a BFMT of a BFSG B of G.
Then (i) BPT(r

−1)=BPT(r) and B
N
T(r
−1)=BNT(r),

(ii) BPT(r)≤B
P
T(e) and B

N
T(r)≥B

N
T(e) for all r,e ∈ G.

Proof. (i) BPT(r
−1) = BP

(α,θ)
(r−1) = αBP(r−1)+ θ = αBP(r)+ θ = BP

(α,θ)
(r). Similarly, BNT(r

−1) =

BNT(r).
(ii) BPT(e)=B

P
(α,θ)
(e)= αBP(e)+θ ≥ αBP(r)+θ =BP

(α,θ)
(r). Similarly, BNT(r)≥B

N
T(e). �

Theorem 3.2. Let M be a BFMT of a BFSG B of G. Then
(i)BPT(ry

−1)=BPT(e)⇒B
P
T(r)=B

P
T(y),

(ii) BNT(ry
−1)=BNT(e)⇒B

N
T(r)=B

N
T(y) for all r,y,e in G.

Proof. (i)

BPT(r)=B
P
(α,θ)(r)

= αBP(r)+θ

= αBP(ry−1y)+θ

≥ αBP(r)+θ

≥ α{min{BP(ry−1),BP(y)}}+θ

≥min{α(BP(ry−1)+θ),α(BP(y)+θ)}



4 Int. J. Anal. Appl. (2022), 20:55

=min{BP(α,θ)(ry
−1),BP(α,θ)(y)}

=min{BP(α,θ)(e),B
P
(α,θ)(y)}

=BP(α,θ)(y).

Similarly, we can prove (ii). �

Theorem 3.3. Let M be a BFMT of a BFSG B of G. Then M is a BFSG of G.

Proof. Let x,y ∈ G.Then we have

BPT(ry
−1)=BP(β,ϑ)(ry

−1)

= βBP(ry−1)+ϑ

≥ βmin{BP(r),BP(y−1}+ϑ

= βmin{BP(r),BP(y)}+ϑ

=min{(β(BP(r)+ϑ)),(β(BP(y)+ϑ))}

=min{BP(β,ϑ)(r),B
P
(β,ϑ)(y)}

=min{BPT(r),B
P
T(y)}.

Theref ore,BPT(ry
−1)≥min{BPT(r),B

P
T(y)}.

Similarly, BNT(ry
−1)≤max{BPT(r), B

P
T(y)}.

Hence, M is a BFSG of G. �

Theorem 3.4. Let M be a BFMT of a BFSG B of G. Then H = {r ∈ G : BPT(r) = B
P
T(e) and

BNT(r)=B
N
T(e)} is a subgroup of G.

Proof. Let r,y ∈ H.
By Theorem 3.1, we have

BPT(r
−1)=BPT(r)=B

P
T(e).

Similarly,

BNT(r
−1)=BNT(r)=B

N
T(e).

Thus BPT(r
−1)=BPT(e) and B

N
T(r
−1)=BNT(e). So r

−1 ∈ H.
Now,

BPT(ry)=B
P
(β,ϑ)(ry)

≥min{BP(β,ϑ)(r),B
P
(β,ϑ)(y)}

=min{BP(β,ϑ)(e),B
P
(β,ϑ)(e)}



Int. J. Anal. Appl. (2022), 20:55 5

=BP(β,ϑ)(e)=B
P
T(e).

Likewise BNT(ry)≤B
N
T(e).

Now,

BPT(e)=B
P
T((ry)(ry)

−1)

geqmin{BPT(ry),B
P
T(ry)}

=BPT(ry).

Similarly, BPT(e)≤B
P
T(ry).

So ry ∈ H.
Hence, H is a subgroup of G. �

The proof of the following two theorems is similar to the proof of Theorem 3.4.

Theorem 3.5. Let M be a BFMT of a BFSG B of G. Then H = {< r,BPT(r) >:B
P
T(r)=B

P
T(e)} is

a fuzzy subgroup (FSG) of G.

Theorem 3.6. Let M be a BFMT of a BFSG B of G. Then H = {< r,BNT(r) >:B
N
T(r)=B

N
T(e)} is

an anti-fuzzy subgroup (AFSG) of G.

Theorem 3.7. Let G and G1 be any two groups. Then the homomorphic image of a BFMT M of a

BFSG B of G is a BFSG of G1.

Proof. Let κ : G → G1 be a homomorphism.
Let V=κ(M),where M is a BFMT of a BFSG B of G.
We shall show that V is a BFSG of G1.
Now, for κ(r) and κ(y) in G1, we have

VP(κ(r)κ(y)−1)=VP(κ(r)κ(y−1))

=VP(κ(ry−1)

≥BPT(ry
−1)

= βBP(ry−1)+α

≥ βmin{BP(r),BP(y−1}+α

= βmin{BP(r),BP(y}+α

=min{(βBP(r)+α),(βBP(y)+α}

≥min{VP(κ(r)),VP(κ(y))}.



6 Int. J. Anal. Appl. (2022), 20:55

Thus VP(κ(r)κ(y)−1)≥min{VP(κ(r)),VP(κ(y))}.
Now,

VN(κ(r)κ(y)−1)=VN(κ(r)κ(y−1))

=VN(κ(ry−1)

≥BNT(ry
−1)

= βBN(ry−1)+α

≤ βmax{BN(r),BN(y−1}+α

= βmax{BN(r),BN(y}+α

=max{(βBN(r)+α),(βBN(y)+α}

≤min{VN(κ(r)),VN(κ(y))}.

Thus VN(κ(r) κ(y)−1)≤max{VN(κ(r)),VN(κ(y))}.

Hence, V is a BFSG of G1.
�

Theorem 3.8. Let G and G1 be any two groups. Then the homomorphic pre-image of a BFMT of a

BFSG B of G1 is a BFSG of G.

Proof. Let G and G1 be any two groups. Let M =κ(B),where M is a BFMT of a BFSG B of G1.
We shall show that B is a BFSG of G.
Now, for κ(r) and κ(y) in G, we have

BP(ry−1)

= BPT(κ(ry
−1))

= BPT(κ(r)κ(y
−1)))

= BPT(κ(r)(κ(y))
−1)

= βBP(κ(r)(κ(y))−1)+α

≥ βmin{BP(κ(r)),BP(κ(y))}+α

= min{βBP(κ(r))+α,βBP(κ(y))+α}

= min{BPT(κ(r)),B
P
T(κ(y))}

= min{BP(κ(r)),BP(κ(y))}

Thus BP(ry−1)≥ min{BP(κ(r)),BP(κ(y))}.



Int. J. Anal. Appl. (2022), 20:55 7

Now,

BN(ry−1)

= BNT (κ(ry
−1))

= BNT (κ(r)κ(y
−1)))

= BNT (κ(r)(κ(y))
−1)

= βBN(κ(r)(κ(y))−1)+α

≤ βmax{BN(κ(r)),BN(κ(y))}+α

= max{βBN(κ(r))+α,βBN(κ(y))+α}

= max{BNT (κ(r)),B
N
T (κ(y))}

= max{BN(κ(r)),BN(κ(y))}

Thus BN(ry−1)≤ max{BN(κ(r)),BN(κ(y))}.
Thus B is a BFSG of G. Thus the homomorphic pre-image of a BFMT of a BFSG B of G1 is a

BFSG of G. �

The proof of the following two theorems is similar to the proof of Theorem 3.7, 3.8.

Theorem 3.9. Let G and G1 be any two groups. Then the homomorphic image of a BFMT M of a

BFNSG B of G is a BFSNG of G1.

Theorem 3.10. Let G and G1 be any two groups. Then the homomorphic pre-image of a BFMT of

a BFNSG B of G1 is a BFNSG of G.

Acknowledgment: The authors wish to thank Prof. K.L.N. Swamy for his valuable suggestions.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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	1. Introduction
	2. Preliminaries
	3. Bipolar fuzzy magnified translations in groups
	References