EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6489
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Tripolar Complex Fuzzy Lie Subalgebras of Lie Algebras

M. Balamurugan1, G. Ellammal1, Aiyared Iampan2,∗

1 Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of
Science and Technology, Chennai 600062, Tamil Nadu, India
2 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang,
Phayao 56000, Thailand

Abstract. The tripolar complex fuzzy set (T CFS) is an extension of the bipolar complex fuzzy
set (BCFS), which itself generalizes traditional fuzzy sets and bipolar fuzzy sets. In this paper, we
further develop this framework by introducing the concept of tripolar complex fuzzy Lie brackets
and investigating their algebraic properties. Additionally, we demonstrate that the scalar multipli-
cation and addition of tripolar complex fuzzy Lie subalgebras yield another tripolar complex fuzzy
Lie subalgebra. Moreover, we establish that the homomorphic image of a nilpotent (or solvable)
tripolar complex fuzzy Lie ideal remains a nilpotent (or solvable) tripolar complex fuzzy Lie ideal.
Finally, we establish that every nilpotent tripolar complex fuzzy Lie ideal is solvable.

2020 Mathematics Subject Classifications: 17B99, 22E60, 08A72, 03E72, 20N25

Key Words and Phrases: Lie algebra, Tripolar complex fuzzy set, Tripolar complex fuzzy Lie
subalgebra, Tripolar complex fuzzy Lie ideal, Nilpotent, Solvable

1. Introduction

A Lie algebra is a fundamental mathematical framework in algebra that has important
applications in many domains, including theoretical physics and mathematics. Lie alge-
bras, named after Norwegian mathematician Sophus Lie [1], provide a robust framework
for examining the algebraic properties of symmetries, vector fields, and transformations.
Over time, academics have thoroughly investigated Lie algebras, revealing a variety of
structural themes that apply to groups, rings, fields, and other algebraic systems. Lie
algebras have applications in domains such as computer science [2] and coding theory [3].

Zadeh’s introduction of fuzzy set theory (FS) in 1965 [4] transformed the mathematical
representation of uncertainty. Zhang introduced bipolar fuzzy sets (BFS) in 1998 [5],
and Lee [6] expanded on this by introducing the notion of bipolar-valued fuzzy sets with
membership functions ξPB̃ : L̃ → [0, 1] and ξNB̃ : L̃ → [−1, 0].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6489

Email addresses: balamurugansvm@gmail.com (M. Balamurugan),
vinoellu@gmail.com (G. Ellammal), aiyared.ia@up.ac.th (A. Iampan)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 2 of 26

In the complex plane, Ramot et al. [7] developed the concept of complex fuzzy sets
(CFS), expanding the range [0, 1] to the unit disk. Tamir et al. [8] expanded on this
notion by mapping the range to [0, 1] + i[0, 1]. Mahmood et al. [9] proposed bipolar
complex fuzzy sets (BCFS), which combine BFS and CFS concepts. Rosenfeld [10] was
the first to connect fuzzy set theory (FS) to group theory. He invented the concept of
fuzzy subgroups and investigated its fundamental features. Following this, other scholars
expanded these ideas to various algebraic structures, including HX-subgroups, ordered
semigroups, and BCK/BCI-algebras, in the setting of uncertainty (see [11–16]).

Yehia [17] introduced the concept of fuzzy Lie algebras by incorporating FS theory
into the study of Lie algebraic ambiguity. Subsequent studies investigated Lie algebras in
the context of FSs (see [18, 19]). Atanassov [20] generalized fuzzy sets by introducing intu-
itionistic fuzzy sets (IFSs), which incorporate degrees of membership, non-membership,
and hesitation to model uncertainty more robustly. Akram and Shum [21] studied intu-
itionistic fuzzy Lie subalgebras and their properties. Akram [22] studied fuzzy Lie ideals
with interval-valued membership functions, as well as solvable and nilpotent Lie L-algebras
utilizing bipolar fuzzy sets (BFS) [23].

Shaqaqha [24] introduced the concept of complex fuzzy Lie subalgebras, examining
their key properties. Kousar et al. [25] studied nilpotent and solvable Lie algebras in an
image fuzzy environment. Al-Masarwah et al. [26] established the notion of crossing cubic
Lie algebras, researching homomorphisms, isomorphism theorems, Cartesian products,
and quotients within this framework. Jaleel et al. [27] proposed interval-valued bipolar
complex fuzzy sets (IVBCFS), whereas Qiyasi et al. [28] created the underlying theory of
confidence-level-based bipolar complex fuzzy sets. Prommai et al. [29] presented tripolar
fuzzy ideals in semigroups. Wattanasiripong et al. [30] introduced tripolar fuzzy pure
ideals in ordered semigroups.

Muhiuddin et al. [14] developed tripolar picture fuzzy ideals of BCK-algebras. Balamu-
rugan et al. [31] introduced complex fuzzy subalgebras and investigated their properties.
Al-Masarwah et al. [32] investigated complex linear diophantine fuzzy ideals in BCK-
algebras. The aim of a tripolar complex fuzzy set (T CFS) is to extend the capabilities of
traditional fuzzy sets, bipolar fuzzy sets, and complex fuzzy sets by incorporating three
independent dimensions of information to model more complex and nuanced real-world
scenarios. From the above literature we found the research gap and tripolar complex
fuzzy set is introduced. The paper is organized as follows: Section 2 provides a concise
set of basic definitions. Section 3 introduces the novel concept of tripolar complex fuzzy
Lie bracket and examines its key properties. Section 4 develops the concept of tripolar
complex fuzzy Lie subalgebras. Section 5 investigates nilpotent and solvable tripolar com-
plex fuzzy Lie ideals. Section 6 concludes the study and suggests promising directions
for future research. To ensure clarity, Table 1 summarizes all mathematical notation and
terminology used throughout the paper.



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 3 of 26

Table 1: Acronyms list.

Acronyms Representations

L Lie Algebra
FS(s) Fuzzy set(s)
CFS(s) Complex fuzzy set(s)
IFS(s) Complex Intuitionistic fuzzy set(s)
BFS(s) Bipolar Fuzzy set(s)
BCFS(s) Bipolar Complex Fuzzy set(s)
T FS(s) Tripolar Fuzzy set(s)
T CFS(s) Tripolar Complex Fuzzy set(s)
T CFLS(s) Tripolar Complex Fuzzy Lie Subalgebra(s)
T CFLI(s) Tripolar Complex Fuzzy Lie Ideal(s)
NT CFLI(s) Nilpotent Tripolar Complex Fuzzy Lie Ideal(s)
ST CFLI(s) Solvable Tripolar Complex Fuzzy Lie Ideal(s)

2. Preliminaries

Definition 1. [21] A Lie algebra L̃ defined over a field F is a vector space together with
a bilinear operation, called the Lie bracket, which adheres to certain key properties:
(i) Bilinear: The Lie bracket [, ] : L̃ × L̃ → L̃ is a bilinear.
(ii) Skew-symmetry: For any ϕ̃, η̃ ∈ L̃,

[ϕ̃, η̃] = −[η̃, ϕ̃].

(iii) Jacobi identity: For any ϕ̃, η̃, ρ̃ ∈ L,

[ϕ̃, [η̃, ρ̃]] + [η̃, [ρ̃, ϕ̃]] + [ρ̃, [ϕ̃, η̃]] = 0.

Definition 2. [4] A FS F̃ = {(ϕ̃, ξF̃ (ϕ̃)) | ϕ̃ ∈ L̃}, where ξF̃ (ϕ̃) : L̃ → [0, 1] is the MD
of ϕ̃ to FS F̃ .

Definition 3. [7] A CFS C̃ in L̃ is the structure

C̃ = {(ϕ̃, ξC̃(ϕ̃) = r̃C̃(ϕ̃)e
i2πω̃C̃(ϕ̃)) | ϕ̃ ∈ L̃},

where r̃C̃(ϕ̃) ∈ [0, 1], ω̃C̃(ϕ̃) ∈ [0, 2π].

Definition 4. [20] An IFS Ĩ in L̃ is the collection Ĩ = {(ϕ̃, ξĨ(ϕ̃), ηĨ(ϕ̃)) | ϕ̃ ∈ L̃},
where ξĨ(ϕ̃) : L̃ → [0, 1] and ηĨ(ϕ̃) : L̃ → [0, 1] denotes the MD and NMD degree of ϕ̃,

respectively with 0 ≤ ξĨ(ϕ̃) + ηĨ(ϕ̃) ≤ 1 for all ϕ̃ ∈ L̃.

Definition 5. [5] A BFS B̃ = {(ϕ̃, ξPB̃ (ϕ̃), ξ
N
B̃ (ϕ̃)) | ϕ̃ ∈ L̃}, where ξPB̃ (ϕ̃) : L̃ → [0, 1] and

ξNB̃ (ϕ̃) : L̃ → [-1, 0] are +MD and −MD of ϕ̃, respectively, with −1 ≤ ξPB̃ (ϕ̃)+ξNB̃ (ϕ̃) ≤ 1.

The pair (µP
B̃ , ξNB̃ ) denotes the BFN . The +MD is the degree of conformity of any

property for an entity, while −MD shows the satisfaction degree of the inherent opposing
characteristic.



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 4 of 26

Definition 6. [9] A BCFS B̃ in L̃ is the structure

B̃ = {(ϕ̃, ξPB̃ (ϕ̃) = r̃PB̃ (ϕ̃)e
i2πω̃P

B̃
(ϕ̃), ξNB̃ (ϕ̃) = r̃NB̃ (ϕ̃)ei2πω̃

N
B̃
(ϕ̃)) | ϕ̃ ∈ L̃},

where ξPB̃ (ϕ̃) : L̃ → [0, 1], ξNB̃ (ϕ̃) : L̃ → [−1, 0].

Definition 7. [29] A T FS ̃ג = {ϕ̃, ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃)} with ϕ̃ ∈ L̃, where ξP̃ג (ϕ̃) : L̃ →

[0, 1] is the +MD, ξÑג (ϕ̃) : L̃ → [−1, 0] is the −MD and ζ̃ג(ϕ̃) : L̃ → [0, 1] is the NM
degree with −1 ≤ ξÑג (ϕ̃) + η̃ג(ϕ̃) ≤ 1 and 0 ≤ ξP̃ג (ϕ̃) + ζ̃ג(ϕ̃) ≤ 1.

Remark 1. A T FS consists of the MD(also called +MD), −MD, and NMD all to-
gether. The +MD(MD) denotes the somewhat satisfied property, NMD denotes the
property that is not fulfilled, and −MD denotes some implicit-counter property (opposite
to MD).

3. Tripolar Complex Fuzzy Bracket Product

Definition 8. A T CFS ̃ג in L̃ is the structure

̃ג = {(ϕ̃, ξP̃ג (ϕ̃) = r̃P̃ג (ϕ̃)e
i2πω̃P

̃ג
(ϕ̃), ξÑג (ϕ̃) = r̃Ñג (ϕ̃)ei2πω̃

N
̃ג
(ϕ̃), ζ̃ג(ϕ̃) = r̃̃ג(ϕ̃)e

i2πω̃̃ג(ϕ̃)) | ϕ̃ ∈ L̃},

where ξP̃ג (ϕ̃) : L̃ → [0, 1], ξÑג (ϕ̃) : L̃ → [−1, 0], ζ̃ג : L̃ → [0, 1].

Example 1. Let L̃ = {ϕ̃1, ϕ̃2, ϕ̃3} be a universe of discourse. A T CFS ̃ג in L̃ is defined
as:

̃ג =



(
ϕ̃1, 0.8e

i2π(0.7),−0.5ei2π(0.3), 0.4ei2π(0.5)
)
,(

ϕ̃2, 0.6e
i2π(0.6),−0.4ei2π(0.4), 0.3ei2π(0.4)

)
,(

ϕ̃3, 0.9e
i2π(0.8),−0.7ei2π(0.2), 0.2ei2π(0.3)

)


Here:

• ξP̃ג (ϕ̃1) = 0.8ei2π(0.7) is the +MD of ϕ̃1 with amplitude 0.8 and phase 0.7,

• ξÑג (ϕ̃1) = −0.5ei2π(0.3) is the −MD of ϕ̃1 with amplitude 0.5 (but sign −1) and
phase 0.3,

• ζ̃ג(ϕ̃1) = 0.4ei2π(0.5) is the NMD of ϕ̃1 with amplitude 0.4 and phase 0.5.

Definition 9. For any T CFSs ̃ג = (ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃)) and ℸ̃ = (ξPℸ̃ (ϕ̃), ξ

N
ℸ̃ (ϕ̃), ζℸ̃(ϕ̃))

of L̃. Then ,̃ג] ℸ̃] = (ξP
,(ϕ̃)[ℸ̃,̃ג] ξ

N
,(ϕ̃)[ℸ̃,̃ג] ζ[̃ג,ℸ̃](ϕ̃)), where

(i) if β̃j ∈ F , ϕ̃j , η̃j ∈ L̃, then

ξP
(ϕ̃)[ℸ̃,̃ג] = supϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{minj∈N {r̃P̃ג (ϕ̃j) ∧ r̃Pℸ̃ (η̃j)}e
i2πminj∈N {ω̃P

̃ג
(ϕ̃j)∧ω̃P

ℸ̃
(η̃j)}



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 5 of 26

and if ϕ̃ ̸=
∑

j∈N βi[ϕ̃i, η̃j ], then ξP
(ϕ̃)[ℸ̃,̃ג] = 0,

(ii) if β̃j ∈ F , ϕ̃j , η̃j ∈ L̃, then

ξN
(ϕ̃)[ℸ̃,̃ג] = inf ϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{maxj∈N {r̃Ñג (ϕ̃j) ∨ r̃Nℸ̃ (η̃j)}ei2πmaxj∈N {ω̃N
̃ג
(ϕ̃j)∨ω̃N

ℸ̃
(η̃j)}

and if ϕ̃ ̸=
∑

j∈N βi[ϕ̃i, η̃j ], then ξN
(ϕ̃)[ℸ̃,̃ג] = 0,

(iii) if β̃j ∈ F , ϕ̃j , η̃j ∈ L̃, then

ζ[̃ג,ℸ̃](ϕ̃) = inf ϕ̃=
∑

j∈N βj [ϕ̃j ,η̃j ]
{maxj∈N {r̃̃ג(ϕ̃j) ∨ r̃ℸ̃(η̃j)}e

i2πmaxj∈N {ω̃̃ג(ϕ̃j)∨ω̃ℸ̃(η̃j)}

and if ϕ̃ ̸=
∑

j∈N βi[ϕ̃i, η̃j ], then ζ[̃ג,ℸ̃](ϕ̃) = 0.

Remark 2. If ϕ̃, η̃ ∈ L̃, then
(1) ξP

,ϕ̃])[ℸ̃,̃ג] η̃]) = r̃P
,ϕ̃])[ℸ̃,̃ג] η̃])e

i2πω̃P
[ℸ̃,̃ג]

([ϕ̃,η̃])
, r̃P

,ϕ̃])[ℸ̃,̃ג] η̃]) ≥ r̃P
r̃∧(ϕ̃)[ℸ̃,̃ג]

P
,(η̃)[ℸ̃,̃ג] and ω̃P

,ϕ̃])[ℸ̃,̃ג] η̃]) ≥
ω̃P
(ϕ̃)[ℸ̃,̃ג] ∧ ω̃P

,(η̃)[ℸ̃,̃ג]

(2) ξN
,ϕ̃])[ℸ̃,̃ג] η̃]) = r̃N

,ϕ̃])[ℸ̃,̃ג] η̃])e
i2πω̃N

[ℸ̃,̃ג]
([ϕ̃,η̃])

, r̃N
,ϕ̃])[ℸ̃,̃ג] η̃]) ≤ r̃N

r̃∨(ϕ̃)[ℸ̃,̃ג]
N
,(η̃)[ℸ̃,̃ג] and ω̃N

,ϕ̃])[ℸ̃,̃ג] η̃]) ≤
ω̃N
(ϕ̃)[ℸ̃,̃ג] ∨ ω̃N

,(η̃)[ℸ̃,̃ג]

(3) ζ[̃ג,ℸ̃]([ϕ̃, η̃]) = r̃[̃ג,ℸ̃]([ϕ̃, η̃])e
i2πω̃[̃ג,ℸ̃]([ϕ̃,η̃]), r̃[̃ג,ℸ̃]([ϕ̃, η̃]) ≤ r̃[̃ג,ℸ̃](ϕ̃)∨r̃[̃ג,ℸ̃](η̃), and ω̃[̃ג,ℸ̃]([ϕ̃, η̃]) ≤

ω̃[̃ג,ℸ̃](ϕ̃) ∨ ω̃N
.(η̃)[ℸ̃,̃ג]

Theorem 1. Let ̃ג = (ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃)), ℸ̃ = (ξPℸ̃ (ϕ̃), ξ

N
ℸ̃ (ϕ̃), ζℸ̃(ϕ̃)), and ℵ̃ = (ξPℵ̃ (ϕ̃), ξ

N
ℵ̃ (ϕ̃), ζℵ̃(ϕ̃))

be T CFSs of L̃ such that ̃ג ⊆ ℵ̃ and ℵ̃ ⊆ ℵ̃. Then +̃ג ℵ̃ ⊆ ℵ̃.

Proof. Let ϕ̃ ∈ L̃. Then

ξP̃ג+ℸ̃(ϕ̃) = r̃P̃ג+ℸ̃(ϕ̃)e
i2πω̃P

ℸ̃+̃ג
(ϕ̃)

≥ sup
ϕ̃ = ϱ̃+ϑ̃

{(r̃P̃ג (ϱ̃) ∧ r̃Pℸ̃ (ϑ̃))e
i2π(ω̃P

̃ג
(ϱ̃)∧ϑ̃P

ℸ̃
(ϑ̃))}

≥ sup
ϕ̃ = ϱ̃+ϑ̃

{(r̃Pℵ̃ (ϱ̃) ∧ r̃Pℵ̃ (ϑ̃))e
i2π(ω̃P

ℵ̃
(ϱ̃)∧ω̃P

ℵ̃
(ϑ̃))}

≥ sup
ϕ̃ = ϱ̃+ϑ̃

{r̃Pℵ̃ (ϱ̃+ ϑ̃)ei2πω̃
P
ℵ̃
(ϱ̃+ϑ̃)}

= r̃Pℵ̃ (ϕ̃)e
i2πω̃P

ℵ̃
(ϕ̃)

= ξPℵ̃ (ϕ̃),

ξÑג+ℸ̃(ϕ̃) = r̃Ñג+ℸ̃(ϕ̃)e
i2πω̃N

ℸ̃+̃ג
(ϕ̃)

≤ inf
ϕ̃ = ϱ̃+ϑ̃

{(r̃Ñג (ϱ̃) ∨ r̃Nℸ̃ (ϑ̃))ei2π(ω̃
N
̃ג
(ϱ̃)∨ω̃N

ℸ̃
(ϑ̃))}



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 6 of 26

≤ inf
ϕ̃ = ϱ̃+ϑ̃

{(r̃Nℵ̃ (ϱ̃) ∨ r̃Nℵ̃ (ϑ̃))ei2π(ω̃
N
ℵ̃
(ϱ̃)∨ω̃N

ℵ̃
(ϑ̃))}

≤ inf
ϕ̃ = ϱ̃+ϑ̃

{r̃Nℵ̃ (ϱ̃+ ϑ̃)ei2πω̃
N
ℵ̃
(ϱ̃+ϑ̃)}

= r̃Nℵ̃ (ϕ̃)ei2πω̃
N
ℵ̃
(ϕ̃)

= ξNℵ̃ (ϕ̃),

and

ζ̃ג+ℸ̃(ϕ̃) = r̃̃ג+ℸ̃(ϕ̃)e
i2πω̃̃ג+ℸ̃(ϕ̃)

≤ inf
ϕ̃ = ϱ̃+ϑ̃

{(r̃̃ג(ϱ̃) ∨ r̃ℸ̃(ϑ̃))e
i2π(ω̃̃ג(ϱ̃)∨ω̃ℸ̃(ϑ̃))}

≤ inf
ϕ̃ = ϱ̃+ϑ̃

{(r̃ℵ̃(ϱ̃) ∨ r̃ℵ̃(ϑ̃))e
i2π(ω̃ℵ̃(ϱ̃)∨ω̃ℵ̃(ϑ̃))}

≤ inf
ϕ̃ = ϱ̃+ϑ̃

{r̃ℵ̃(ϱ̃+ ϑ̃)ei2πω̃ℵ̃(ϱ̃+ϑ̃)}

= r̃ℵ̃(ϕ̃)e
i2πω̃ℵ̃(ϕ̃)

= ζℵ̃(ϕ̃).

Hence, +̃ג ℸ̃ ⊆ ℵ̃.

Theorem 2. Let 1̃ג = (ξP1̃ג
(ϕ̃), ξN1̃ג

(ϕ̃), ζ1̃ג(ϕ̃)), 2̃ג = (ξP2̃ג
(ϕ̃), ξN2̃ג

(ϕ̃), ζ2̃ג(ϕ̃)) and ℸ̃1 =

(ξPℸ̃1
(ϕ̃), ξNℸ̃1

(ϕ̃), ζℸ̃1
(ϕ̃)), ℸ̃2 = (ξPℸ̃2

(ϕ̃), ξNℸ̃2
(ϕ̃), ζℸ̃2

(ϕ̃)) be T CFSs of L̃ such that 1̃ג ⊆
,2̃ג ℸ̃1 ⊆ ℸ̃2. Then ,1̃ג] ℸ̃] ⊆ ,2̃ג] ℸ̃] and ,̃ג] ℸ̃1] ⊆ ,̃ג] ℸ̃2].

Proof. Let ϕ̃ ∈ L̃. Then

ξP
[ℸ̃1,1̃ג]

(ϕ̃) = sup
ϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{min
j∈N

{r̃P1̃ג(ϕ̃j) ∧ r̃Pℸ̃1
(η̃j)}e

i2πmaxj∈N {ω̃P
1̃ג
(ϕ̃j)∧ω̃P

ℸ̃1
(η̃j)}}

≥ sup
ϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{min
j∈N

{r̃P2̃ג(ϕ̃j) ∧ r̃Pℸ̃2
(η̃j)}e

i2πminj∈N {ω̃P
2̃ג
(ϕ̃j)∧ω̃P

ℸ̃2
(η̃j)}}

= ξP
[ℸ̃2,2̃ג]

(ϕ̃),

ξN
[ℸ̃1,1̃ג]

(ϕ̃) = inf
ϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{max
j∈N

{r̃N1̃ג (ϕ̃j) ∨ r̃Nℸ̃1
(η̃j)}e

i2πmaxj∈N {ω̃N
1̃ג
(ϕ̃j)∨ω̃N

ℸ̃1
(η̃j)}}

≤ inf
ϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{max
j∈N

{r̃N2̃ג (ϕ̃j) ∨ r̃Nℸ̃2
(η̃j)}e

i2πmaxj∈N {ω̃N
2̃ג
(ϕ̃j)∨ω̃N

ℸ̃2
(η̃j)}}

= ξN
[ℸ̃2,2̃ג]

(ϕ̃),



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 7 of 26

and

ζ[1̃ג,ℸ̃1]
(ϕ̃) = inf

ϕ̃=
∑

j∈N βj [ϕ̃j ,η̃j ]
{max
j∈N

{r̃1̃ג(ϕ̃j) ∨ r̃ℸ̃1
(η̃j)}ei2πmaxj∈N {ω̃1̃ג

(ϕ̃j)∨ω̃ℸ̃1
(η̃j)}}

≤ inf
ϕ̃=

∑
j∈N βj [ϕ̃j ,η̃j ]

{max
j∈N

{r̃2̃ג(ϕ̃j) ∨ r̃ℸ̃2
(η̃j)}ei2πmaxj∈N {ω̃2̃ג

(ϕ̃j)∨ω̃ℸ̃2
(η̃j)}}

= ζ[2̃ג,ℸ̃2]
(ϕ̃).

Theorem 3. Let 1̃ג = (ξP1̃ג
(ϕ̃), ξN1̃ג

(ϕ̃), ζ1̃ג(ϕ̃)), 2̃ג = (ξP2̃ג
(ϕ̃), ξN2̃ג

(ϕ̃), ζ2̃ג(ϕ̃)) and ℸ̃1 =

(ξPℸ̃1
(ϕ̃), ξNℸ̃1

(ϕ̃), ζℸ̃1
(ϕ̃)), ℸ̃2 = (ξPℸ̃2

(ϕ̃), ξNℸ̃2
(ϕ̃), ζℸ̃2

(ϕ̃)) and ̃ג = (ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃)), ℸ̃ =

(ξPℸ̃ (ϕ̃), ξ
N
ℸ̃ (ϕ̃), ζℸ̃(ϕ̃)) be T CFSs of L̃. Then +1̃ג] ,2̃ג ℸ̃] = ,1̃ג] ℸ̃]+[2̃ג, ℸ̃] and ,̃ג] ℸ̃1+ℸ̃2] =

,̃ג] ℸ̃1] + ,̃ג] ℸ̃2].

Proof. Let ϕ̃ ∈ L̃ and sup = supϕ̃=
∑

j∈N βj [ϕ̃j ,η̃j ]
. Then

ξP
[ℸ̃,2̃ג+1̃ג]

(ϕ̃)

= supϕ̃=
∑

j∈N βj [ϕ̃j ,η̃j ]
{minj∈N {r̃P2̃ג+1̃ג

(ϕ̃j) ∧ r̃Pℸ̃ (η̃j)}e
i2πminj∈N {ω̃P

2̃ג+1̃ג
(ϕ̃j)∧ω̃P

ℸ̃
(η̃j)}}

= sup{minj∈N {supϕ̃j=ϱ̃j+ϑ̃j
{r̃P1̃ג(ϱ̃j)∧r̃

P
2̃ג
(ϑ̃j)}∧r̃Pℸ̃ (η̃j)}e

i2πminj∈N {supϕ̃j=ϱ̃j+ϑ̃j
{ω̃P

1̃ג
(ϱ̃j)∧ω̃P

2̃ג
(ϑ̃j)}∧ω̃P

ℸ̃
(η̃j)}}

= sup{minj∈N {supϕ̃j=ϱ̃j+ϑ̃j
{r̃P1̃ג(ϱ̃j)∧r̃

P
2̃ג
(ϑ̃j)∧r̃Pℸ̃ (η̃j)}}e

i2πminj∈N {supϕ̃j=ϱ̃j+ϑ̃j
{ω̃P

1̃ג
(ϱ̃j)∧ω̃P

2̃ג
(ϑ̃j)∧ω̃P

ℸ̃
(η̃j)}}}

= sup{minj∈N {supϕ̃j=ϱ̃j+ϑ̃j
{r̃P1̃ג(ϱ̃j)∧r̃

P
2̃ג
(ϑ̃j)∧r̃Pℸ̃ (η̃j)}}}e

i2πsup{minj∈N {supϕ̃j=ϱ̃j+ϑ̃j
{ω̃P

1̃ג
(ϱ̃j)∧ω̃P

2̃ג
(ϑ̃j)∧ω̃P

ℸ̃
(η̃j)}}}

= r̃P
[ℸ̃,2̃ג+1̃ג]

(ϕ̃)e
i2πω̃P

[ℸ̃,2̃ג+1̃ג]
(ϕ̃)

≥ r̃P
[ℸ̃,2̃ג]+[ℸ̃,1̃ג]

(ϕ̃)e
i2πω̃P

[ℸ̃,2̃ג]+[ℸ̃,1̃ג]
(ϕ̃)

≥ ξP
[ℸ̃,2̃ג]+[ℸ̃,1̃ג]

(ϕ̃),

ξN
[ℸ̃,2̃ג+1̃ג]

(ϕ̃)

= inf ϕ̃=
∑

j∈N βj [ϕ̃j ,η̃j ]
{maxj∈N {r̃N2̃ג+1̃ג

(ϕ̃j) ∨ r̃Nℸ̃ (η̃j)}e
i2πmaxj∈N {ω̃N

2̃ג+1̃ג
(ϕ̃j)∨ω̃N

ℸ̃
(η̃j)}}

= inf{maxj∈N {inf ϕ̃j=ϱ̃j+ϑ̃j
{r̃N1̃ג (ϱ̃j)∨r̃

N
2̃ג
(ϑ̃j)}∨r̃Nℸ̃ (η̃j)}e

i2πmaxj∈N {infϕ̃j=ϱ̃j+ϑ̃j
{ω̃N

1̃ג
(ϱ̃j)∨ω̃N

2̃ג
(ϑ̃j)}∨ω̃N

ℸ̃
(η̃j)}}

= inf{maxj∈N {inf ϕ̃j=ϱ̃j+ϑ̃j
{r̃N1̃ג (ϱ̃j)∨r̃

N
2̃ג
(ϑ̃j)∨r̃Nℸ̃ (η̃j)}}e

i2πmaxj∈N {infϕ̃j=ϱ̃j+ϑ̃j
{ω̃N

1̃ג
(ϱ̃j)∨ω̃N

2̃ג
(ϑ̃j)∨ω̃N

ℸ̃
(η̃j)}}}

= inf{maxj∈N {inf ϕ̃j=ϱ̃j+ϑ̃j
{r̃N1̃ג (ϱ̃j)∨r̃

N
2̃ג
(ϑ̃j)∨r̃Nℸ̃ (η̃j)}}}e

i2πinf{maxj∈N {infϕ̃j=ϱ̃j+ϑ̃j
{ω̃N

1̃ג
(ϱ̃j)∨ω̃N

2̃ג
(ϑ̃j)∨ω̃N

ℸ̃
(η̃j)}}}

= r̃N
[ℸ̃,2̃ג+1̃ג]

(ϕ̃)e
i2πω̃N

[ℸ̃,2̃ג+1̃ג]
(ϕ̃)

≤ r̃N
[ℸ̃,2̃ג]+[ℸ̃,1̃ג]

(ϕ̃)e
i2πω̃N

[ℸ̃,2̃ג]+[ℸ̃,1̃ג]
(ϕ̃)

≤ ξN
[ℸ̃,2̃ג]+[ℸ̃,1̃ג]

(ϕ̃),

and
ζ[2̃ג+1̃ג,ℸ̃](ϕ̃)



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 8 of 26

= inf ϕ̃=
∑

j∈N βj [ϕ̃j ,η̃j ]
{maxj∈N {r̃2̃ג+1̃ג(ϕ̃j) ∨ r̃ℸ̃(η̃j)}e

i2πmaxj∈N {ω̃2̃ג+1̃ג
(ϕ̃j)∨ω̃ℸ̃(η̃j)}}

= inf{maxj∈N {inf ϕ̃j=ϱ̃j+ϑ̃j
{r̃1̃ג(ϱ̃j)∨r̃2̃ג(ϑ̃j)}∨r̃ℸ̃(η̃j)}e

i2πmaxj∈N {infϕ̃j=ϱ̃j+ϑ̃j
{ω̃N

1̃ג
(ϱ̃j)∨ω̃2̃ג

(ϑ̃j)}∨ω̃ℸ̃(η̃j)}}

= inf{maxj∈N {inf ϕ̃j=ϱ̃j+ϑ̃j
{r̃1̃ג(ϱ̃j)∨r̃2̃ג(ϑ̃j)∨r̃ℸ̃(η̃j)}}e

i2πmaxj∈N {infϕ̃j=ϱ̃j+ϑ̃j
{ω̃N

1̃ג
(ϱ̃j)∨ω̃2̃ג

(ϑ̃j)∨ω̃ℸ̃(η̃j)}}}

= inf{maxj∈N {inf ϕ̃j=ϱ̃j+ϑ̃j
{r̃1̃ג(ϱ̃j)∨r̃2̃ג(ϑ̃j)∨r̃ℸ̃(η̃j)}}}e

i2πinf{maxj∈N {infϕ̃j=ϱ̃j+ϑ̃j
{ω̃1̃ג

(ϱ̃j)∨ω̃2̃ג
(ϑ̃j)∨ω̃ℸ̃(η̃j)}}}

= r̃[2̃ג+1̃ג,ℸ̃](ϕ̃)e
i2πω̃[2̃ג+1̃ג,ℸ̃](ϕ̃)

≤ r̃[1̃ג,ℸ̃]+[2̃ג,ℸ̃](ϕ̃)e
i2πω̃[1̃ג,ℸ̃]+[2̃ג,ℸ̃](ϕ̃)

≤ ζ[1̃ג,ℸ̃]+[2̃ג,ℸ̃](ϕ̃).

This shows that 1̃ג] + ,2̃ג ℸ̃] ⊆ ,1̃ג] ℸ̃] + ,2̃ג] ℸ̃]. Let ϕ̃ ∈ L̃. Then

ξP2̃ג+1̃ג
(ϕ̃) = r̃P2̃ג+1̃ג

(ϕ̃)e
i2πω̃P

2̃ג+1̃ג
(ϕ̃)

= sup
ϕ̃ = ϱ̃+ϑ̃

{(r̃P1̃ג(ϱ̃) ∧ r̃P2̃ג
(ϑ̃))e

i2π(ω̃P
1̃ג
(ϱ̃)∧ω̃P

2̃ג
(ϑ̃))}

≥ (r̃P1̃ג
(ϕ̃) ∧ r̃P2̃ג

(0))e
i2π(ω̃P

1̃ג
(ϕ̃)∧ω̃P

2̃ג
(0))

= r̃P1̃ג
(ϕ̃)e

i2πω̃P
1̃ג
(ϕ̃)

,

ξN2̃ג+1̃ג
(ϕ̃) = r̃N2̃ג+1̃ג

(ϕ̃)e
i2πω̃N

2̃ג+1̃ג
(ϕ̃)

= inf
ϕ̃ = ϱ̃+ϑ̃

{(r̃N1̃ג (ϱ̃) ∨ r̃N2̃ג
(ϑ̃))e

i2π(ω̃N
1̃ג
(ϱ̃)∨ω̃N

2̃ג
(ϑ̃))}

≤ (r̃N1̃ג
(ϕ̃) ∨ r̃N2̃ג

(0))e
i2π(ω̃N

1̃ג
(ϕ̃)∨ω̃N

2̃ג
(0))

= r̃N1̃ג
(ϕ̃)e

i2πω̃N
1̃ג
(ϕ̃)

,

and

ζ (ϕ̃)2̃ג+1̃ג = r̃2̃ג+1̃ג(ϕ̃)e
i2πω̃2̃ג+1̃ג

(ϕ̃)

= inf
ϕ̃ = ϱ̃+ϑ̃

{(r̃1̃ג(ϱ̃) ∨ r̃2̃ג(ϑ̃))e
i2π(ω̃1̃ג

(ϱ̃)∨ω̃2̃ג
(ϑ̃))}

≤ (r̃1̃ג(ϕ̃) ∨ r̃(0)2̃ג)e
i2π(ω̃1̃ג

(ϕ̃)∨ω̃2̃ג
(0))

= r̃1̃ג(ϕ̃)e
i2πω̃1̃ג

(ϕ̃)
.

Therefore, 1̃ג ⊆ .2̃ג+1̃ג Similarly, 2̃ג ⊆ .2̃ג+1̃ג By Theorem 2, we have ,1̃ג] ℸ̃] ⊆ ,2̃ג+1̃ג] ℸ̃],
,1̃ג] ℸ̃] ⊆ +1̃ג] ,2̃ג ℸ̃]. So, by Theorem 1, ,1̃ג] ℸ̃] + ,2̃ג] ℸ̃] ⊆ +1̃ג] ,2̃ג ℸ̃]. Hence, +1̃ג] ,2̃ג ℸ̃] =
,1̃ג] ℸ̃] + ,2̃ג] ℸ̃].



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 9 of 26

Theorem 4. Let ̃ג = (ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃)), ℸ̃ = (ξPℸ̃ (ϕ̃), ξ

N
ℸ̃ (ϕ̃), ζℸ̃(ϕ̃)) be T CFSs of L̃.

Then

[β̃̃ג, ℸ̃] = β̃[̃ג, ℸ̃] and ,̃ג] β̃ℸ̃] = β̃[̃ג, ℸ̃],

for every β̃ ∈ F .

Proof. Let ϕ̃ ∈ L̃. Then

ξP
[β̃̃ג,ℸ̃](ϕ̃) = sup

ϕ̃=
∑

j∈N β̃j [ϕ̃j ,η̃j ]

{min
j∈N

{r̃P
β̃̃ג(ϕ̃j) ∧ r̃Pℸ̃ (η̃j)}e

i2πminj∈N {ω̃P
β̃̃ג

(ϕ̃j)∧ω̃P
ℸ̃
(η̃j)}}

= sup
ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{min
j∈N

{r̃P̃ג (β̃
−1ϕ̃j) ∧ r̃Pℸ̃ (η̃j)}e

i2πminj∈N {ω̃P
̃ג
(β̃−1ϕ̃j)∧ω̃P

ℸ̃
(η̃j)}}

= sup
ϕ̃=

∑
j∈N β̃β̃j [β̃−1ϕ̃j ,η̃j ]

{min
j∈N

{r̃P̃ג (β̃
−1ϕ̃j) ∧ r̃Pℸ̃ (η̃j)}e

i2πminj∈N {ω̃P
̃ג
(β̃−1ϕ̃j)∧ω̃P

ℸ̃
(η̃j)}}

= ξP
β̃)[ℸ̃,̃ג]

−1ϕ̃)

= ξP
β̃[̃ג,ℸ̃](ϕ̃),

ξN
[β̃̃ג,ℸ̃](ϕ̃) = inf

ϕ̃=
∑

j∈N β̃j [ϕ̃j ,η̃j ]
{max
j∈N

{r̃N
β̃̃ג(ϕ̃j) ∨ r̃Nℸ̃ (η̃j)}e

i2πmaxj∈N {ω̃N
β̃̃ג

(ϕ̃j)∨ω̃N
ℸ̃
(η̃j)}}

= inf
ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{max
j∈N

{r̃Ñג (β̃−1ϕ̃j) ∨ r̃Nℸ̃ (η̃j)}ei2πmaxj∈N {ω̃N
̃ג
(β̃−1ϕ̃j)∨ω̃N

ℸ̃
(η̃j)}}

= inf
ϕ̃=

∑
j∈N β̃β̃j [β̃−1ϕ̃j ,η̃j ]

{max
j∈N

{r̃Ñג (β̃−1ϕ̃j) ∨ r̃Nℸ̃ (η̃j)}ei2πmaxj∈N {ω̃N
̃ג
(β̃−1ϕ̃j)∨ω̃N

ℸ̃
(η̃j)}}

= ξN
β̃)[ℸ̃,̃ג]

−1ϕ̃)

= ξN
β̃[̃ג,ℸ̃](ϕ̃),

and

ζ[β̃̃ג,ℸ̃](ϕ̃) = inf
ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{max
j∈N

{r̃β̃̃ג(ϕ̃j) ∨ r̃ℸ̃(η̃j)}e
i2πmaxj∈N {ω̃β̃̃ג(ϕ̃j)∨ω̃ℸ̃(η̃j)}}

= inf
ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{max
j∈N

{r̃̃ג(β̃
−1ϕ̃j) ∨ r̃ℸ̃(η̃j)}e

i2πmaxj∈N {ω̃̃ג(β̃
−1ϕ̃j)∨ω̃ℸ̃(η̃j)}}

= inf
ϕ̃=

∑
j∈N β̃β̃j [β̃−1ϕ̃j ,η̃j ]

{max
j∈N

{r̃̃ג(β̃
−1ϕ̃j) ∨ r̃ℸ̃(η̃j)}e

i2πmaxj∈N {ω̃̃ג(β̃
−1ϕ̃j)∨ω̃ℸ̃(η̃j)}}

= ξ[̃ג,ℸ̃](β̃
−1ϕ̃)

= ξβ̃[̃ג,ℸ̃](ϕ̃).

If β̃ = 0, ϕ̃ ̸= 0, then

ξP
[β̃̃ג,ℸ̃](ϕ̃) = sup

ϕ̃=
∑

j∈N β̃j [ϕ̃j ,η̃j ]

{min
j∈N

{r̃P
β̃̃ג(ϕ̃j) ∧ r̃Pℸ̃ (η̃j)}e

i2πminj∈N {ω̃P
β̃̃ג

(ϕ̃j)∧ω̃P
ℸ̃
(η̃j)}},



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 10 of 26

ξN
[β̃̃ג,ℸ̃](ϕ̃) = inf

ϕ̃=
∑

j∈N β̃j [ϕ̃j ,η̃j ]
{max
j∈N

{r̃N
β̃̃ג(ϕ̃j) ∨ r̃Nℸ̃ (η̃j)}e

i2πmaxj∈N {ω̃N
β̃̃ג

(ϕ̃j)∨ω̃N
ℸ̃
(η̃j)}},

and

ζ[β̃̃ג,ℸ̃](ϕ̃) = inf
ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{max
j∈N

{r̃β̃̃ג(ϕ̃j) ∨ r̃ℸ̃(η̃j)}e
i2πmaxj∈N {ω̃β̃̃ג(ϕ̃j)∨ω̃ℸ̃(η̃j)}},

there exists ϕ̃ ̸= 0, which implies that r̃P
β̃̃ג(ϕ̃j) = 0, ω̃P

β̃̃ג(ϕ̃j) = 0, r̃N
β̃̃ג(ϕ̃j) = 1, ω̃N

β̃̃ג(ϕ̃j) = 1,

and r̃β̃̃ג(ϕ̃j) = 1, ω̃β̃̃ג(ϕ̃j) = 1. So, ξP
[β̃̃ג,ℸ̃](ϕ̃) = 0, ξN

[β̃̃ג,ℸ̃](ϕ̃) = 1 and ζ[β̃̃ג,ℸ̃](ϕ̃) = 1. If β̃ = 0

and ϕ̃ = 0, then it is obvious. The second one can be obtained in the same way.

4. Tripolar Complex Fuzzy Lie Subalgebras

Definition 10. A T CFS ̃ג = {(ϕ̃, ξP̃ג (ϕ̃) = r̃P̃ג (ϕ̃)e
i2πω̃P

̃ג
(ϕ̃), ξÑג (ϕ̃) = r̃Ñג (ϕ̃)ei2πω̃

P
̃ג
(ϕ̃), ζ̃ג(ϕ̃) =

r̃̃ג(ϕ̃)e
i2πω̃̃ג(ϕ̃)) | ϕ̃ ∈ L̃} on L̃ is termed as a T CFLS on L̃ if ϕ̃, η̃ ∈ L̃, β̃ ∈ F , the under-

neath holds:
(i) ξP̃ג (ϕ̃ + η̃) ≥ ξP̃ג (ϕ̃) ∧ ξP̃ג (η̃) ⇒ r̃P̃ג (ϕ̃ + η̃)ei2πω̃

P
̃ג
(ϕ̃+η̃) ≥ r̃P̃ג (ϕ̃ + η̃)ei2πω̃

P
̃ג
(ϕ̃) ∧ r̃P̃ג (ϕ̃ +

η̃)ei2πω̃
P
̃ג
(η̃),

(ii) ξÑג (ϕ̃+ η̃) ≤ ξÑג (ϕ̃) ∨ ξÑג (η̃) ⇒ r̃Ñג (ϕ̃+ η̃)ei2πω̃
N
̃ג
(ϕ̃+η̃) ≤ r̃Ñג (ϕ̃+ η̃)ei2πω̃

N
̃ג
(ϕ̃) ∨ r̃Ñג (ϕ̃+

η̃)ei2πω̃
N
̃ג
(η̃),

(iii) ζ̃ג(ϕ̃+ η̃) ≤ ζ̃ג(ϕ̃)∨ζ̃ג(η̃) ⇒ r̃̃ג(ϕ̃+ η̃)ei2πω̃̃ג(ϕ̃+η̃) ≤ r̃̃ג(ϕ̃+ η̃)ei2πω̃̃ג(ϕ̃)∨ r̃̃ג(ϕ̃+ η̃)ei2πω̃̃ג(η̃),

(iv) ξP̃ג (β̃ϕ̃) ≥ ξP̃ג (ϕ̃) ⇒ r̃P̃ג (β̃ϕ̃)e
i2πω̃P

̃ג
(β̃ϕ̃) ≥ r̃P̃ג (ϕ̃)e

i2πω̃P
̃ג
(ϕ̃),

(ϑ̃) ξÑג (β̃ϕ̃) ≤ ξÑג (ϕ̃) ⇒ r̃Ñג (β̃ϕ̃)ei2πω̃
N
̃ג
(β̃ϕ̃) ≤ r̃Ñג (ϕ̃)ei2πω̃

N
̃ג
(ϕ̃),

(vi) ζ̃ג(β̃ϕ̃) ≤ ζ̃ג(ϕ̃) ⇒ r̃̃ג(β̃ϕ̃)e
i2πω̃̃ג(β̃ϕ̃) ≤ r̃̃ג(ϕ̃)e

i2πω̃̃ג(ϕ̃),

(vii) ξP̃ג ([ϕ̃, η̃]) ≥ ξP̃ג (ϕ̃)∧ ξP̃ג (η̃) ⇒ r̃P̃ג ([ϕ̃, η̃])e
i2πω̃P

̃ג
([ϕ̃,η̃]) ≥ r̃P̃ג (ϕ̃)e

i2πω̃P
̃ג
(ϕ̃) ∧ r̃̃ג(η̃)e

i2πω̃P
̃ג
(η̃),

(viii) ξÑג ([ϕ̃, η̃]) ≤ ξÑג (ϕ̃)∨ξÑג (η̃) ⇒ r̃Ñג ([ϕ̃, η̃])ei2πω̃
N
̃ג
([ϕ̃,η̃]) ≥ r̃Ñג (ϕ̃)ei2πω̃

N
̃ג
(ϕ̃)∨r̃Ñג (η̃)ei2πω̃

N
̃ג
(η̃),

(ix) ζ̃ג([ϕ̃, η̃]) ≤ ζ̃ג(ϕ̃) ∨ ζ̃ג(η̃) ⇒ r̃̃ג([ϕ̃, η̃])e
i2πω̃̃ג([ϕ̃,η̃]) ≥ r̃̃ג(ϕ̃)e

i2πω̃̃ג(ϕ̃) ∨ r̃̃ג(η̃)e
i2πω̃̃ג(η̃).

When the conditions (vii), (viii) and (ix) are modified to

(x) ξP̃ג ([ϕ̃, η̃]) ≥ ξP̃ג (ϕ̃) ∨ ξP̃ג (η̃) ⇒ r̃P̃ג ([ϕ̃, η̃])e
i2πω̃P

̃ג
([ϕ̃,η̃]) ≥ r̃P̃ג (ϕ̃)e

i2πω̃P
̃ג
(ϕ̃) ∨ r̃̃ג(η̃)e

i2πω̃P
̃ג
(η̃),

(xi) ξÑג ([ϕ̃, η̃]) ≤ ξÑג (ϕ̃)∧ξÑג (η̃) ⇒ r̃Ñג ([ϕ̃, η̃])ei2πω̃
N
̃ג
([ϕ̃,η̃]) ≥ r̃Ñג (ϕ̃)ei2πω̃

N
̃ג
(ϕ̃)∧r̃Ñג (η̃)ei2πω̃

N
̃ג
(η̃),

(xii) ζ̃ג([ϕ̃, η̃]) ≤ ζ̃ג(ϕ̃) ∧ ζ̃ג(η̃) ⇒ r̃̃ג([ϕ̃, η̃])e
i2πω̃̃ג([ϕ̃,η̃]) ≥ r̃̃ג(ϕ̃)e

i2πω̃̃ג(ϕ̃) ∧ r̃̃ג(η̃)e
i2πω̃̃ג(η̃), the

T CFS ̃ג is a tripolar complex fuzzy Lie ideal (shortly, T CFLI) on L̃.

Definition 11. Let ̃ג = {(ϕ̃, ξP̃ג (ϕ̃) = r̃P̃ג (ϕ̃)e
i2πω̃P

̃ג
(ϕ̃), ξÑג (ϕ̃) = r̃Ñג (ϕ̃)ei2πω̃

P
̃ג
(ϕ̃), ζ̃ג(ϕ̃) =

r̃̃ג(ϕ̃)e
i2πω̃̃ג(ϕ̃)) | ϕ̃ ∈ L̃} be a T CFS on L̃. For β̃ ∈ F and ϕ̃ ∈ L̃, define β̃̃ג =

{(ϕ̃, ξP
β̃̃ג(ϕ̃), ξ

N
β̃Ñ (ϕ̃), ζβ̃̃ג(ϕ̃)) | ϕ̃ ∈ L̃}, where



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 11 of 26

ξP
β̃̃ג(ϕ̃) =


ξP̃ג (β̃

−1ϕ̃) if β̃ ̸= 0

1 if β̃ = 0, ϕ̃ = 0

0 if β̃ = 0, ϕ̃ ̸= 0,

ξN
β̃̃ג(ϕ̃) =


ξÑג (β̃−1ϕ̃) if β̃ ̸= 0

0 if β̃ = 0, ϕ̃ = 0

1 if β̃ = 0, ϕ̃ ̸= 0,

and

ζβ̃̃ג(ϕ̃) =


ζ̃ג(β̃

−1ϕ̃) if β̃ ̸= 0

0 if β̃ = 0, ϕ̃ = 0

1 if β̃ = 0, ϕ̃ ̸= 0.

Example 2. Let L̃ = {ϕ̃, η̃, ρ̃} and define the T CFS ̃ג on L̃ as:

̃ג =


(ϕ̃, 0.5ei2π(0.2), 0.4ei2π(0.2), 0.6ei2π(0.3)),

(η̃, 1ei2π(0), 0ei2π(0), 0ei2π(0)),

(ρ̃, 0.8ei2π(0.1), 0.2ei2π(0.1), 0.7ei2π(0.25))


Let β̃ = 1. Then

ξP
β̃̃ג(ϕ̃) = ξP̃ג (β̃

−1ϕ̃) = ξP̃ג (ϕ̃)

and similarly for ξN and ζ. Therefore,

β̃̃ג = ̃ג (since β̃ = 1)

Now consider β̃ = 0. Then for each ϕ̃ ∈ L̃,

ξP
β̃̃ג(ϕ̃) =

{
1 if ϕ̃ = 0

0 if ϕ̃ ̸= 0
, ξN

β̃̃ג(ϕ̃) =

{
0 if ϕ̃ = 0

1 if ϕ̃ ̸= 0
,

ζβ̃̃ג(ϕ̃) =

{
0 if ϕ̃ = 0

1 if ϕ̃ ̸= 0

So,

β̃̃ג =

 (ϕ̃, 0, 1, 1),
(η̃, 1, 0, 0),
(ρ̃, 0, 1, 1)


Theorem 5. Let ̃ג be a T CFLS on .̃ג Then β̃̃ג is also a T CFLS for any β̃ ∈ F .

Proof. Let ̃ג = {(ϕ̃, ξP̃ג (ϕ̃) = r̃P̃ג (ϕ̃)e
i2πω̃P

̃ג
(ϕ̃), ξÑג (ϕ̃) = r̃Ñג (ϕ̃)ei2πω̃

P
̃ג
(ϕ̃), ζ̃ג(ϕ̃) = r̃̃ג(ϕ̃)e

i2πω̃̃ג(ϕ̃)) |
ϕ̃ ∈ L̃} be a T CFLS on L̃. Define β̃̃ג = {(ϕ̃, ξP

β̃̃ג(ϕ̃), ξ
N
β̃Ñ (ϕ̃), ζβ̃̃ג(ϕ̃)) | ϕ̃ ∈ L̃}, where

ξP
β̃̃ג(ϕ̃) = ξP̃ג (β̃

−1ϕ̃),



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 12 of 26

ξN
β̃̃ג(ϕ̃) = ξÑג (β̃−1ϕ̃),

ζβ̃̃ג(ϕ̃) = ζ̃ג(β̃
−1ϕ̃).

We need to verify that β̃̃ג is also a T CFLS for any β̃ ∈ F . Let ϕ̃, η̃ ∈ L̃.
Case 1.

ξP
β̃̃ג(ϕ̃+ η̃) = ξP̃ג (β̃

−1(ϕ̃+ η̃))

= ξP̃ג (β̃
−1ϕ̃+ β̃−1η̃)

≥ ξP̃ג (β̃
−1ϕ̃) ∧ ξP̃ג (β̃

−1η̃)

= ξP
β̃̃ג(ϕ̃) ∧ ξP

β̃̃ג(η̃).

Case 2.

ξN
β̃̃ג(ϕ̃+ η̃) = ξÑג (β̃−1(ϕ̃+ η̃))

= ξÑג (β̃−1ϕ̃+ β̃−1η̃)

≤ ξÑג (β̃−1ϕ̃) ∨ ξÑג (β̃−1η̃)

= ξN
β̃̃ג(ϕ̃) ∨ ξN

β̃̃ג(η̃).

Case 3.

ζβ̃̃ג(ϕ̃+ η̃) = ζ̃ג(β̃
−1(ϕ̃+ η̃))

= ζ̃ג(β̃
−1ϕ̃+ β̃−1η̃)

≤ ζ̃ג(β̃
−1ϕ̃) ∨ ζ̃ג(β̃

−1η̃)

= ζβ̃̃ג(ϕ̃) ∨ ζβ̃̃ג(η̃).

Case 4.

ξP
β̃̃ג(β̃ϕ̃) = ξP̃ג (β̃

−1β̃ϕ̃)

= ξP̃ג (β̃(β̃
−1ϕ̃))

≥ ξP̃ג (β̃
−1ϕ̃)

= ξP
β̃̃ג(ϕ̃).

Case 5.

ξN
β̃̃ג(β̃ϕ̃) = ξÑג (β̃−1β̃ϕ̃)

= ξÑג (β̃(β̃−1ϕ̃))

≤ ξÑג (β̃−1ϕ̃)

= ξN
β̃̃ג(ϕ̃).



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 13 of 26

Case 6.

ζβ̃̃ג(β̃ϕ̃) = ζ̃ג(β̃
−1β̃ϕ̃)

= ζ̃ג(β̃(β̃
−1ϕ̃))

≤ ζ̃ג(β̃
−1ϕ̃)

= ζβ̃̃ג(ϕ̃).

Case 7.

ξP
β̃̃ג([ϕ̃, η̃]) = ξP̃ג (β̃

−1[ϕ̃, η̃])

= ξP̃ג ([β̃
−1ϕ̃, β̃−1η̃])

≥ ξP̃ג (β̃
−1ϕ̃) ∧ ξP̃ג (β̃

−1η̃)

= ξP
β̃̃ג(ϕ̃) ∧ ξP

β̃̃ג(η̃).

Case 8.

ξN
β̃̃ג([ϕ̃, η̃]) = ξÑג (β̃−1[ϕ̃, η̃])

= ξÑג ([β̃−1ϕ̃, β̃−1η̃])

≤ ξÑג (β̃−1ϕ̃) ∨ ξÑג (β̃−1η̃)

= ξN
β̃̃ג(ϕ̃) ∨ ξN

β̃̃ג(η̃).

Case 9.

ζβ̃̃ג([ϕ̃, η̃]) = ζ̃ג(β̃
−1[ϕ̃, η̃])

= ζ̃ג([β̃
−1ϕ̃, β̃−1η̃])

≤ ζ̃ג(β̃
−1ϕ̃) ∨ ξÑג (β̃−1η̃)

= ζβ̃̃ג(ϕ̃) ∨ ζβ̃̃ג(η̃).

Hence, β̃̃ג is indeed a T CFLS on L̃.

Definition 12. Let ̃ג = {(ϕ̃, ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃))} and ℸ̃ = {(ϕ̃, ξPℸ̃ (ϕ̃), ξ

N
ℸ̃ (ϕ̃), ζℸ̃(ϕ̃))} be

two T CFLS on L̃. Then +̃ג ℸ̃ = {(ϕ̃, ξP̃ג+ℸ̃(ϕ̃), ξ
N
,ℸ̃(ϕ̃)+̃ג ζ̃ג+ℸ̃(ϕ̃))} be T CFS on L̃ given by

ξP̃ג+ℸ̃(ϕ̃) =

{
supϕ̃=ε̃+ς̃

{
ξP̃ג (ε̃) ∧ ξPℸ̃ (ς̃)

}
if ϕ̃ = ε̃+ ς̃

0 otherwise,

ξÑג+ℸ̃(ϕ̃) =

{
inf ϕ̃=ε̃+ς̃

{
ξÑג (ε̃) ∨ ξNℸ̃ (ς̃)

}
if ϕ̃ = ε̃+ ς̃

1 otherwise,

and



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 14 of 26

ζ̃ג+ℸ̃(ϕ̃) =

{
inf ϕ̃=ε̃+ς̃

{
ζ̃ג(ε̃) ∨ ζℸ̃(ς̃)

}
if ϕ̃ = ε̃+ ς̃

1 otherwise.

Theorem 6. Let ̃ג and ℸ̃ be a T CFLS of L̃. Then +̃ג ℸ̃ is a T CFLS of L̃.

Proof. Let ̃ג = {(ϕ̃, ξP̃ג (ϕ̃), ξ
N
̃ג (ϕ̃), ζ̃ג(ϕ̃))} and ℸ̃ = {(ϕ̃, ξPℸ̃ (ϕ̃), ξ

N
ℸ̃ (ϕ̃), ζℸ̃(ϕ̃))} be two

T CFLS on L̃. Define their sum as:

+̃ג ℸ̃ = {(ϕ̃, ξP̃ג+ℸ̃(ϕ̃), ξ
N
,ℸ̃(ϕ̃)+̃ג ζ̃ג+ℸ̃(ϕ̃))},

where for each ϕ̃ ∈ L̃,

ξP̃ג+ℸ̃(ϕ̃) = sup
ϕ̃=ε̃+ς̃

{
ξP̃ג (ε̃) ∧ ξPℸ̃ (ς̃)

}
ξÑג+ℸ̃(ϕ̃) = inf

ϕ̃=ε̃+ς̃

{
ξÑג (ε̃) ∨ ξNℸ̃ (ς̃)

}
ζ̃ג+ℸ̃(ϕ̃) = inf

ϕ̃=ε̃+ς̃

{
ζ̃ג(ε̃) ∨ ζℸ̃(ς̃)

}
To verify that +̃ג ℸ̃ is a T CFLS of L̃. Let ϕ̃, η̃ ∈ L̃.
Case 1.

ξP̃ג+ℸ̃(ϕ̃+ η̃) = sup
(ϕ̃+η̃)=ε̃+ς̃

{
ξP̃ג (ε̃) ∧ ξPℸ̃ (ς̃)

}
≥ sup

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
ξP̃ג (ε̃1 + ε̃2) ∧ ξPℸ̃ (ς̃1 + ς̃2)

}
≥ sup

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
(ξP̃ג (ε̃1) ∧ ξP̃ג (ε̃2)) ∧ (ξPℸ̃ (ς̃1) ∧ ξPℸ̃ (ς̃2))

}

=

(
sup

ϕ̃=ε̃1+ς̃1

{ξP̃ג (ε̃1) ∧ ξPℸ̃ (ς̃1)}

)
∧

(
sup

η̃=ε̃2+ς̃2

{ξP̃ג (ε̃2) ∧ ξPℸ̃ (ς̃2)}

)
= ξP̃ג+ℸ̃(ϕ̃) ∧ ξP̃ג+ℸ̃(η̃).

Case 2.

ξÑג+ℸ̃(ϕ̃+ η̃) = inf
(ϕ̃+η̃)=ε̃+ς̃

{
ξÑג (ε̃) ∨ ξNℸ̃ (ς̃)

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
ξÑג (ε̃1 + ε̃2) ∨ ξNℸ̃ (ς̃1 + ς̃2)

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
(ξÑג (ε̃1) ∨ ξÑג (ε̃2)) ∨ (ξNℸ̃ (ς̃1) ∨ ξNℸ̃ (ς̃2))

}

=

(
inf

ϕ̃=ε̃1+ς̃1

{ξÑג (ε̃1) ∨ ξNℸ̃ (ς̃1)}

)
∨
(

inf
η̃=ε̃2+ς̃2

{ξÑג (ε̃2) ∨ ξNℸ̃ (ς̃2)}
)



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 15 of 26

= ξÑג+ℸ̃(ϕ̃) ∨ ξÑג+ℸ̃(η̃).

Case 3.

ζ̃ג+ℸ̃(ϕ̃+ η̃) = inf
(ϕ̃+η̃)=ε̃+ς̃

{
ζ̃ג(ε̃) ∨ ζℸ̃(ς̃)

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
ζ̃ג(ε̃1 + ε̃2) ∨ ζℸ̃(ς̃1 + ς̃2)

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
(ζ̃ג(ε̃1) ∨ ζ̃ג(ε̃2)) ∨ (ζℸ̃(ς̃1) ∨ ζℸ̃(ς̃2))

}

=

(
inf

ϕ̃=ε̃1+ς̃1

{ζ̃ג(ε̃1) ∨ ζℸ̃(ς̃1)}

)
∨
(

inf
η̃=ε̃2+ς̃2

{ζ̃ג(ε̃2) ∨ ζℸ̃(ς̃2)}
)

= ζ̃ג+ℸ̃(ϕ̃) ∨ ζ̃ג+ℸ̃(η̃).

Case 4.

ξP̃ג+ℸ̃(β̃ϕ̃) = sup
β̃ϕ̃=ε̃+ς̃

{
ξP̃ג (ε̃) ∧ ξPℸ̃ (ς̃)

}
= sup

ϕ̃=ε̃/β̃+ς̃/β̃

{
ξP̃ג (β̃(ε̃/β̃)) ∧ ξPℸ̃ (β̃(ς̃/β̃))

}
≥ sup

ϕ̃=ε̃′+ς̃′

{
ξP̃ג (ε̃

′) ∧ ξPℸ̃ (ς̃
′)
}

= ξP̃ג+ℸ̃(ϕ̃).

Case 5.

ξÑג+ℸ̃(β̃ϕ̃) = inf
β̃ϕ̃=ε̃+ς̃

{
ξÑג (ε̃) ∨ ξNℸ̃ (ς̃)

}
= inf

ϕ̃=ε̃/β̃+ς̃/β̃

{
ξÑג (β̃(ε̃/β̃)) ∨ ξNℸ̃ (β̃(ς̃/β̃))

}
≤ inf

ϕ̃=ε̃′+ς̃′

{
ξÑג (ε̃′) ∨ ξNℸ̃ (ς̃ ′)

}
= ξÑג+ℸ̃(ϕ̃)

Case 6.

ζ̃ג+ℸ̃(β̃ϕ̃) = inf
β̃ϕ̃=ε̃+ς̃

{
ζ̃ג(ε̃) ∨ ζℸ̃(ς̃)

}
= inf

ϕ̃=ε̃/β̃+ς̃/β̃

{
ζ̃ג(β̃(ε̃/β̃)) ∨ ζℸ̃(β̃(ς̃/β̃))

}
≤ inf

ϕ̃=ε̃′+ς̃′

{
ζ̃ג(ε̃

′) ∨ ζℸ̃(ς̃
′)
}

= ζ̃ג+ℸ̃(ϕ̃).



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 16 of 26

Case 7.

ξP̃ג+ℸ̃([ϕ̃, η̃]) = sup
[ϕ̃,η̃]=ε̃+ς̃

{
ξP̃ג (ε̃) ∧ ξPℸ̃ (ς̃)

}
≥ sup

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
ξP̃ג ([ε̃1, ε̃2]) ∧ ξPℸ̃ ([ς̃1, ς̃2])

}
≥ sup

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
(ξP̃ג (ε̃1) ∧ ξP̃ג (ε̃2)) ∧ (ξPℸ̃ (ς̃1) ∧ ξPℸ̃ (ς̃2))

}
= ξP̃ג+ℸ̃(ϕ̃) ∧ ξP̃ג+ℸ̃(η̃).

Case 8.

ξÑג+ℸ̃([ϕ̃, η̃]) = inf
[ϕ̃,η̃]=ε̃+ς̃

{
ξÑג (ε̃) ∨ ξNℸ̃ (ς̃)

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
ξÑג ([ε̃1, ε̃2]) ∨ ξNℸ̃ ([ς̃1, ς̃2])

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
(ξÑג (ε̃1) ∨ ξÑג (ε̃2)) ∨ (ξNℸ̃ (ς̃1) ∨ ξNℸ̃ (ς̃2))

}
= ξÑג+ℸ̃(ϕ̃) ∨ ξÑג+ℸ̃(η̃).

Case 9.

ζ̃ג+ℸ̃([ϕ̃, η̃]) = inf
[ϕ̃,η̃]=ε̃+ς̃

{
ζ̃ג(ε̃) ∨ ζℸ̃(ς̃)

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
ζ̃ג([ε̃1, ε̃2]) ∨ ζℸ̃([ς̃1, ς̃2])

}
≤ inf

ϕ̃=ε̃1+ς̃1
η̃=ε̃2+ς̃2

{
(ζ̃ג(ε̃1) ∨ ζ̃ג(ε̃2)) ∨ (ζℸ̃(ς̃1) ∨ ζℸ̃(ς̃2))

}
= ζ̃ג+ℸ̃(ϕ̃) ∨ ζ̃ג+ℸ̃(η̃).

Hence, +̃ג ℸ̃ is indeed a +̃ג ℸ̃ is a T CFLS of L̃.

5. Nilpotent and Solvable Tripolar Complex Fuzzy Lie Ideals

Definition 13. A T CFLI ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃̃ג) is called nilpotent if there exists a positive

integer n such that ñג = 0.

Theorem 7. A homomorphic image of an NT CFLI is an NT CFLI.

Proof. Let f : L̃1 → L̃2 be a homomorphism of L̃, and let ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃̃ג) be an

NT CFLI in L̃1. Assume f(̃ג) = ℸ̃. We proceed by induction to establish that f(̃ג(n)) ⊇



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 17 of 26

ℸ̃(n) for n ∈ Ñ . As a base case, we first demonstrate that f([̃ג, ([̃ג ⊇ [f(̃ג), f(̃ג)] = [ℸ̃, ℸ̃].
Let η̃ ∈ L̃2. Then

f
(
⟨⟨ξP̃ג , ξ

P
̃ג ⟩⟩
)
(η̃) = sup

{
⟨⟨ξP̃ג , ξ

P
̃ג ⟩⟩(ϕ̃) | f(ϕ̃) = η̃

}
= sup

{
sup

{
min

{
ξP̃ג (ε̃), ξ

P
̃ג (ς̃)

}
| ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃

}}
= sup

{
min

{
ξP̃ג (ε̃), ξ

P
̃ג (ς̃)

}
| ε̃, ς̃ ∈ L̃, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃

}
= sup

{
min

{
ξP̃ג (ε̃), ξ

P
̃ג (ς̃)

}
| ε̃, ς̃ ∈ L̃1, [f(ε̃), f(ς̃)] = η̃

}
= sup

{
min

{
ξP̃ג (ε̃), ξ

P
̃ג (ς̃)

}
| ε̃, ς̃ ∈ L̃1, f(ε̃) = ϖ̃, f(ς̃) = ϑ̃, [ϖ̃, ϑ̃] = η̃

}
≥ sup

{
min

{
sup

ε̃∈f−1(ϖ̃)

ξP̃ג (ε̃), sup
ς̃∈f−1(ϑ̃)

ξP̃ג (ς̃)

}
| [ϖ̃, ϑ̃] = η̃

}
= sup

{
min

{
f(ξP̃ג (ϖ̃)), f(ξP̃ג (ϑ̃))

}
| [ϖ̃, ϑ̃] = η̃

}
= ⟨⟨f(ξP̃ג ), f(ξ

P
̃ג )⟩⟩(η̃)

and

f
(
⟨⟨ξÑג , ξÑג ⟩⟩

)
(η̃) = inf

{
⟨⟨ξÑג , ξÑג ⟩⟩(ϕ̃) | f(ϕ̃) = η̃

}
= inf

{
inf
{
max

{
ξÑג (ε̃), ξÑג (ς̃)

}
| ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃

}}
= inf

{
max

{
ξÑג (ε̃), ξÑג (ς̃)

}
| ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃

}
= inf

{
max

{
ξÑג (ε̃), ξÑג (ς̃)

}
| ε̃, ς̃ ∈ L̃1, [f(ε̃), f(ς̃)] = η̃

}
= inf

{
max

{
ξÑג (ε̃), ξÑג (ς̃)

}
| f(ε̃) = ϖ̃, f(ς̃) = ϑ̃, [ϖ̃, ϑ̃] = η̃

}
≤ inf

{
max

{
inf

ε̃∈f−1(ϖ̃)
ξÑג (ε̃), inf

ς̃∈f−1(ϑ̃)
ξÑג (ς̃)

}
| [ϖ̃, ϑ̃] = η̃

}
= inf

{
max

{
f(ξÑג (ϖ̃)), f(ξÑג (ϑ̃))

}
| [ϖ̃, ϑ̃] = η̃

}
= ⟨⟨f(ξÑג ), f(ξÑג )⟩⟩(η̃).

Also,

f
(
⟨⟨ζ̃ג, ζ̃ג⟩⟩

)
(η̃) = inf

{
⟨⟨ζ̃ג, ζ̃ג⟩⟩(ϕ̃) | f(ϕ̃) = η̃

}
= inf

{
inf
{
max

{
ζ̃ג(ε̃), ζ̃ג(ς̃)

}
| ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃

}}
= inf

{
max

{
ζ̃ג(ε̃), ζ̃ג(ς̃)

}
| ε̃, ς̃ ∈ L̃1, [ε̃, ς̃] = ϕ̃, f(ϕ̃) = η̃

}
= inf

{
max

{
ζ̃ג(ε̃), ζ̃ג(ς̃)

}
| ε̃, ς̃ ∈ L̃1, [f(ε̃), f(ς̃)] = η̃

}



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 18 of 26

= inf
{
max

{
ζ̃ג(ε̃), ζ̃ג(ς̃)

}
| f(ε̃) = ϖ̃, f(ς̃) = ϑ̃, [ϖ̃, ϑ̃] = η̃

}
≤ inf

{
max

{
inf

ε̃∈f−1(ϖ̃)
ζ̃ג(ε̃), inf

ς̃∈f−1(ϑ̃)
ζ̃ג(ς̃)

}
| [ϖ̃, ϑ̃] = η̃

}
= inf

{
max

{
f(ζ̃ג(ϖ̃)), f(ζ̃ג(ϑ̃))

}
| [ϖ̃, ϑ̃] = η̃

}
= ⟨⟨f(ζ̃ג), f(ζ̃ג)⟩⟩(η̃).

Thus,

f([̃ג, ([̃ג ⊇ f(⟨⟨̃ג, (⟨⟨̃ג ⊇ ⟨⟨f(̃ג), f(̃ג)⟩⟩ = [f(̃ג), f(̃ג)].

For n > 1, we get

f(̃גn) = f(̃ג · (n−1̃ג ⊇ [f(̃ג), f(̃גn−1)] ⊇ [ℸ̃, ℸ̃n−1] = ℸ̃n.

Let m be a positive integer such that m̃ג = 0. Then, for 0 ̸= η̃ ∈ L̃2,

ξPℸ̃(m)(η̃) ≤ f(ξP̃גn)(η̃) = f(0)η̃ = sup {0(ε̃) | f(ε̃) = η̃} = 0,

ξNℸ̃m(η) ≤ f(ξÑגn)(η̃) = f(0)η̃ = inf {1(ε̃) | f(ε̃) = η̃} = 0.

Also,

ζ̃ℸ̃m(η̃) ≥ f(ζ̃̃גn)(η̃) = inf {1(ε̃) | f(ε̃) = η̃} = 0.

Thus, ℸ̃m = 0.

Definition 14. Let ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃̃ג) and ℸ̃ = (ξPℸ̃ , ξ

N
ℸ̃ , ζ̃ℸ̃) be two T CFLI of L̃. The sum

⊕̃ג ℸ̃ is called a direct sum if ̃ג ∩ ℸ̃ = 0.

Theorem 8. The direct sum of two NT CFLIs is also an NT CFLI.

Proof. Suppose that ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃̃ג) and ℸ̃ = (ξPℸ̃ , ξ

N
ℸ̃ , ζ̃ℸ̃) are two T CFLIs such that

̃ג ∩ ℸ̃ = 0. We prove that ̃ג ∩ ℸ̃ = 0. Let ϕ̃(̸= 0) ∈ L̃. Then

⟨⟨ξP̃ג , ξ
P
ℸ̃ ⟩⟩(ϕ̃) = sup

{
min

{
ξP̃ג (ε̃), ξ

P
ℸ̃ (ϑ̃)

}
| [ε̃, ς̃] = ϕ̃

}
≤ min

{
ξP̃ג (ϕ̃), ξ

P
ℸ̃ (ϕ̃)

}
= 0

and

⟨⟨ξÑג , ξNℸ̃ ⟩⟩(ϕ) = inf
{
max

{
ξÑג (ε̃), ξNℸ̃ (ϑ̃)

}
| [ε̃, ς̃] = ϕ̃

}
≥ max

{
ξÑג (ϕ̃), ξNℸ̃ (ϕ̃)

}



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 19 of 26

= 0.

Also,

⟨⟨ζ̃ג, ζℸ̃⟩⟩(ϕ) = inf
{
max

{
ζ̃ג(ε̃), ζℸ̃(ϑ̃)

}
| [ε̃, ς̃] = ϕ̃

}
≥ max

{
ζ̃ג(ϕ̃), ζℸ̃(ϕ̃)

}
= 0.

Therefore, ̃ג ∩ ℸ̃ = 0 ⇒ ,m̃ג] ℸ̃n] = 0, for all positive integers m,n. Also, we claim that
⊕̃ג) ℸ̃)n ⊆ ñג ⊕ ℸ̃n for n ∈ Ñ . We proceed by induction on n. For n = 1,

⊕̃ג) ℸ̃)1 = ⊕̃ג] ℸ̃, ⊕̃ג ℸ̃] ⊆ ,̃ג] ⊕[̃ג ,̃ג] ℸ̃]⊕ [ℸ̃, ⊕[̃ג [ℸ̃, ℸ̃] = 1̃ג + ℸ̃1.

Now, for n > 1,

⊕̃ג) ℸ̃)n = ⊕̃ג] ℸ̃, ⊕̃ג) ℸ̃)n−1]

⊆ ⊕̃ג] ℸ̃, n−1̃ג ⊕ ℸ̃n−1]

⊆ ,̃ג] ⊕[n−1̃ג ,̃ג] ℸ̃n−1]⊕ [ℸ̃, ⊕[n−1̃ג [ℸ̃, ℸ̃n−1]

= ñג ⊕ ℸ̃n.

Since, there are two positive integers r&t such that r̃ג = ℸ̃t = 0, we have

⊕̃ג) ℸ̃)r+t ⊆ r+t̃ג ⊕ ℸ̃r+t = 0.

Definition 15. A T CFLI ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃̃ג) is called solvable if there exists a positive

integer n such that ℸ̃(n) = 0.

Theorem 9. An NT CFLI is solvable.

Proof. To establish the desired result, it suffices to show that (n)̃ג ⊆ ,ñג for all positive
integer n. We will prove this statement by induction on n, making use of Theorem 2,

(1)̃ג = ,̃ג] [̃ג = (1)̃ג

(2)̃ג = ,(1)̃ג] [(1)̃ג ⊆ ,̃ג] [(1)̃ג = 2̃ג

(3)̃ג = ,(2)̃ג] [(2)̃ג ⊆ ,̃ג] [(2)̃ג = 3̃ג

...

(n)̃ג = ,(n−1)̃ג] [(n−1)̃ג ⊆ ,̃ג] [(n−1)̃ג ⊆ ,̃ג] [(n−1)̃ג = .ñג

Definition 16. Let ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃ג) be a T CFLI of L̃. Construct a sequence of T CFLIs

of L̃ by



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 20 of 26

(0)̃ג = ,̃ג (1)̃ג = ,(0)̃ג] ,[(0)̃ג (2)̃ג = ,(1)̃ג] ,[(1)̃ג ..., (n)̃ג = ,(n−1)̃ג] ,[(n−1)̃ג

then (n)̃ג is called the nth derived TCFLI of L̃. In which, (i+1)̃ג = (ξP̃ג(i+1) , ξ
N
(i+1)̃ג , ζ̃ג(i+1)),

where
(i) if β̃j ∈ F , ϕ̃j , η̃j ∈ L̃, then
ξP̃ג(i+1)(ϕ̃) = supϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{minj∈N {r̃P̃ג(i)(ϕ̃j) ∧ r̃P̃ג(i)(η̃j)}e
i2πminj∈N {ω̃P

(i)̃ג
(ϕ̃j)∧ω̃P

(i)̃ג
(η̃j)}}

and if ϕ̃ ̸=
∑

j∈N β̃i[ϕ̃i, η̃j ], then ξP̃ג(i+1)(ϕ̃) = 0,

(ii) if β̃j ∈ F , ϕ̃j , η̃j ∈ L̃, then
ξÑג(i+1)(ϕ̃) = inf ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{maxj∈N {r̃Ñג(i)(ϕ̃j) ∨ r̃Ñג(i)(η̃j)}e
i2πmaxj∈N {ω̃N

(i)̃ג
(ϕ̃j)∨ω̃N

(i)̃ג
(η̃j)}}

and if ϕ̃ ̸=
∑

j∈N β̃i[ϕ̃i, η̃j ], then ξÑג(i+1)(ϕ̃) = 0,

(iii) if β̃j ∈ F , ϕ̃j , η̃j ∈ L̃, then
ζ̃ג(i+1)(ϕ̃) = inf ϕ̃=

∑
j∈N β̃j [ϕ̃j ,η̃j ]

{maxj∈N {r̃̃ג(i)(ϕ̃j) ∨ r̃̃ג(i)(η̃j)}e
i2πmaxj∈N {ω̃̃ג(i) (ϕ̃j)∨ω̃̃ג(i) (η̃j)}}

and if ϕ̃ ̸=
∑

j∈N β̃i[ϕ̃i, η̃j ], then ζ̃ג(i+1)(ϕ̃) = 0.

Remark 3. From the Definition 13, we can get ξP(0)̃ג ⊇ ξP(1)̃ג ⊇ ξP(2)̃ג ⊇ ... ⊇ ξP̃ג(n) ⊇ ...,

ξN(0)̃ג ⊆ ξN(1)̃ג ⊆ ξN(2)̃ג ⊆ ... ⊆ ξÑג(n) ⊆ ..., and ζ (0)̃ג ⊆ ζ (1)̃ג ⊆ ζ (2)̃ג ⊆ ... ⊆ ζ (n)̃ג ⊆ ....

Definition 17. A ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃ג) be a T CFLI of L̃. Then ̃ג = (ξP̃ג , ξ

N
̃ג , ζ̃ג) is ST CFLI

if and if there exists a positive integer n such that ξP̃ג(m)(n) = 10, ξÑגm(n) = (−1)0 and
ζ̃גm(n) = 00, for all positive integer m ≥ n.

Theorem 10. A homomorphic images of ST CFLIs are ST CFLIs.

Proof. Consider f : L̃ → L̃′
be a homomorphism of L̃, and assume that ̃ג = (ξP̃ג , ξ

N
̃ג , ζ̃ג)

is a T CFLI L̃. Let f(̃ג) = ℸ̃, i.e., ξPℸ̃ = ξP
f(̃ג), ξNℸ̃ = ξN

f(̃ג). We aim to prove, by induction

on n, that ξP
f(̃ג(n))

= ξPℸ̃(n) and ξN
f(̃ג(n))

= ξNℸ̃(n) , for all positive integer n. As the base case,

consider n = 1. Let η̃ ∈ L̃′
. Then

ξP
f((1)̃ג)(η̃)

= ξP
f([T̃ ,T̃ ])

(η̃)

= sup
η̃=f(ϕ̃)

{
ξP
[T̃ ,T̃ ]

(η̃)
}

= sup
η̃=f(ϕ̃)

 sup
ϕ̃=

∑
j∈N

β̃j [ϕ̃j ,η̃j ]

{
min
j∈N

{
r̃P̃ג (ϕ̃j) ∧ r̃P̃ג (η̃j)

}
ei2πminj∈N {ω̃P

̃ג
(ϕ̃j)∧ω̃P

̃ג
(η̃j)}

}
= sup

ϕ̃=
∑

j∈N
β̃j [ϕ̃j ,η̃j ]

{
min
j∈N

{
r̃P̃ג (ϕ̃j) ∧ r̃P̃ג (η̃j)

}
ei2πminj∈N {ω̃P

̃ג
(ϕ̃j)∧ω̃P

̃ג
(η̃j)}

}

= sup
ϕ̃=

∑
j∈N

β̃j [ε̃j ,ς̃j ]

{
min
j∈N

{
r̃P̃ג (ϕ̃j) ∧ r̃P̃ג (η̃j)

}
ei2πminj∈N {ω̃P

̃ג
(ϕ̃j)∧ω̃P

̃ג
(η̃j)} | f(ϕ̃j) = ε̃j , f(η̃j) = ς̃j

}



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 21 of 26

= sup∑
j∈N

β̃j [ε̃j ,ς̃j ]=ϕ̃

{
min
j∈N

{
r̃Pℸ̃ (ε̃j) ∧ r̃Pℸ̃ (ς̃j)

}
ei2πminj∈N {ω̃P

ℸ̃
(ε̃j)∧ω̃P

ℸ̃
(ς̃j)}

}

= ξP
[ℸ̃,ℸ̃](η̃)

= ξPℸ̃(1)(η̃)

and

ξN
f((1)̃ג)(η̃)

= ξN
f([T̃ ,T̃ ])

(η̃)

= inf
η̃=f(ϕ̃)

{
ξN
[T̃ ,T̃ ]

(η̃)
}

= inf
η̃=f(ϕ̃)

 inf
ϕ̃=

∑
j∈N

β̃j [ϕ̃j ,η̃j ]

{
max
j∈N

{
r̃Ñג (ϕ̃j) ∧ r̃Ñג (η̃j)

}
ei2πmaxj∈N {ω̃N

̃ג
(ϕ̃j)∧ω̃N

̃ג
(η̃j)}

}
= inf

ϕ̃=
∑

j∈N
β̃j [ϕ̃j ,η̃j ]

{
max
j∈N

{
r̃Ñג (ϕ̃j) ∧ r̃Ñג (η̃j)

}
ei2πmaxj∈N {ω̃N

̃ג
(ϕ̃j)∧ω̃N

̃ג
(η̃j)}

}

= inf
ϕ̃=

∑
j∈N

β̃j [ε̃j ,ς̃j ]

{
max
j∈N

{
r̃Ñג (ϕ̃j) ∧ r̃Ñג (η̃j)

}
ei2πmaxj∈N {ω̃N

̃ג
(ϕ̃j)∧ω̃N

̃ג
(η̃j)} | f(ϕ̃j) = ε̃j , f(η̃j) = ς̃j

}

= inf∑
j∈N

β̃j [ε̃j ,ς̃j ]=ϕ̃

{
max
j∈N

{
r̃Nℸ̃ (ε̃j) ∧ r̃Nℸ̃ (ς̃j)

}
ei2πmaxj∈N {ω̃N

ℸ̃
(ε̃j)∧ω̃N

ℸ̃
(ς̃j)}

}
= ξN

[ℸ̃,ℸ̃](η̃)

= ξNℸ̃(1)(η̃).

Also,

ζf((1)̃ג)(η̃)

= ζf([T̃ ,T̃ ])(η̃)

= inf
η̃=f(ϕ̃)

{
ζN
[T̃ ,T̃ ]

(η̃)
}

= inf
η̃=f(ϕ̃)

 inf
ϕ̃=

∑
j∈N

β̃j [ϕ̃j ,η̃j ]

{
max
j∈N

{
r̃̃ג(ϕ̃j) ∧ r̃̃ג(η̃j)

}
ei2πmaxj∈N {ω̃̃ג (ϕ̃j)∧ω̃̃ג (η̃j)}

}
= inf

ϕ̃=
∑

j∈N
β̃j [ϕ̃j ,η̃j ]

{
max
j∈N

{
r̃̃ג(ϕ̃j) ∧ r̃̃ג(η̃j)

}
ei2πmaxj∈N {ω̃̃ג (ϕ̃j)∧ω̃̃ג (η̃j)}

}



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 22 of 26

= inf
ϕ̃=

∑
j∈N

β̃j [ε̃j ,ς̃j ]

{
max
j∈N

{
r̃̃ג(ϕ̃j) ∧ r̃̃ג(η̃j)

}
ei2πmaxj∈N {ω̃̃ג (ϕ̃j)∧ω̃̃ג (η̃j)} | f(ϕ̃j) = ε̃j , f(η̃j) = ς̃j

}

= inf∑
j∈N

β̃j [ε̃j ,ς̃j ]=ϕ̃

{
max
j∈N

{
r̃ℸ̃(ε̃j) ∧ r̃ℸ̃(ς̃j)

}
ei2πmaxj∈N {ω̃ℸ̃ (ε̃j)∧ω̃ℸ̃ (ς̃j)}

}
= ζ[ℸ̃,ℸ̃](η̃)

= ζℸ̃(1)(η̃).

The statement holds for the base case n = 1. Now, assume that it is true for some positive
integer n− 1; that is, we assume the induction hypothesis holds for n− 1. Then

ξP
f(̃ג(n))

= ξP
f([̃ג(n−1),̃ג(n−1)])

= ξP
[f(̃ג(n−1)),f(̃ג(n−1))]

= ξP
[ℸ̃(n−1),ℸ̃(n−1)]

= ξPℸ̃(n) ,

ξN
f(̃ג(n))

= ξN
f([̃ג(n−1),̃ג(n−1)])

= ξN
[f(̃ג(n−1)),f(̃ג(n−1))]

= ξN
[ℸ̃(n−1),ℸ̃(n−1)]

= ξNℸ̃(n)

and

ζf(̃ג(n)) = ζf([̃ג(n−1),̃ג(n−1)]) = ζ[f(̃ג(n−1)),f(̃ג(n−1))] = ζ[ℸ̃(n−1),ℸ̃(n−1)] = ζℸ̃(n) .

Let ξP̃ג(m) = 10, ξ
N
(m)̃ג = (−1)0, and ζ̃ג(m) = 00. Then

ξPℸ̃(m)(η̃) = ξP
f(̃גm)

(η̃) = sup
η̃=f(ϕ̃)

{
10(ϕ̃)

}
= 0,

ξNℸ̃(m)(η̃) = ξN
f(̃גm)

(η̃) = inf
η̃=f(ϕ̃)

{
(−1)0(ϕ̃)

}
= 0,

and

ζℸ̃(m)(η̃) = ζf(̃גm)(η̃) = inf
η̃=f(ϕ̃)

{
00(ϕ̃)

}
= 0,

for every 0 ̸= η̃ ∈ L̃′ . So ξPℸ̃(m) = 10, ξ
N
ℸ̃(m) = (−1)0 and ζℸ̃(m) = 00.

Theorem 11. Let ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃) be a T CFLI of L̃ and suppose that the quotient ̃ג

J̃
forms a ST CFLI in the quotient algebra L̃

J̃ . Assume that ℸ̃ = (ξPℸ̃ , ξ
N
ℸ̃ , ζ̃ℸ̃) is a ST CFLI

of L̃, and also serves as a T CFLI of ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃). If, in addition, ℸ̃(J̃ ) = J̃)̃ג ), then

̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃̃ג) is solvable.

Proof. Let f : L̃ → L̃
J̃ be the canonical map. According to the argument presented in

the proof of Theorem 7,

ξP
ϕ(̃ג(n))

= ξP
( ̃ג
J )(n)

, ξN
ϕ(̃ג(n))

= ξP
( ̃ג
J )(n)

and ζϕ(̃ג(n)) = ζ
( ̃ג
J )(n) .



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 23 of 26

Since ̃ג
J is solvable, there exists a positive integer n such that ξP

( ̃ג
J̃
)(n)

= 10, ξ
N
( ̃ג
J̃
)(n)

= (−1)0

and ζ
( ̃ג
J̃
)(n) = 00.

For 0 ̸= η̃ ∈ L̃
J̃ , we have

sup
m∈f−1(η̃)

{
ξP̃ג(n)(m)

}
= ξP

f(̃ג(n))
(η̃) = ξP

( ̃ג
J̃
)(n)

(η̃) = 0,

inf
m∈f−1(η̃)

{
ξÑג(n)(m)

}
= ξN

f(̃ג(n))
(η̃) = ξN

( ̃ג
J̃
)(n)

(η̃) = 0,

and

inf
m∈f−1(η̃)

{
ζ̃ג(n)(m)

}
= ζf(̃ג(n))(η̃) = ζ

( ̃ג
J̃
)(n)(η̃) = 0.

Note that m ̸= 0 and m ∈ L̃. Then ξP̃ג(n)(m) = 0, ξÑג(n)(m) = 0 and ζ̃ג(n)(m) = 0.
For η̃ = 0, we have

sup
m∈f−1(0)

{
ξP
((n)̃ג)

(m)
}
= ξP

f(̃ג(n))
(0) = 1,

inf
m∈f−1(0)

{
ξN
((n)̃ג)

(m)
}
= ξN

f(̃ג(n))
(0) = −1

and

inf
m∈f−1(0)

{
ζ(̃ג(n))(m)

}
= ζf(̃ג(n))(0) = 0.

Since f−1(0) = J̃ and ℸ̃(J̃ ) = J̃)̃ג ), we have

ξPℸ̃(n)(J̃ ) = ξP̃ג(n)(J̃ ), ξNℸ̃(n)(J̃ ) = ξÑג(n)(J̃ ) and ζℸ̃(n)(J̃ ) = ζ̃ג(n)(J̃ ).

For any ϕ̃ ∈ J̃ , ℸ̃ is solvable, then there exists a positive integer n such that ξP
s̃(n) = 10

and ξN
s̃(n) = (−1)0,

ξPℸ̃(n) = 10, ξ
N
(n)̃ג = (−1)0 and ζ̃ג(n) = 00.

Hence, ξP̃ג(n) = 10, ξÑג(n) = (−1)0 and ζ̃ג(n) = 00, which implies that ̃ג = (ξP̃ג , ξ
N
̃ג , ζ̃) is

solvable.

6. Conclusion

In this paper, we extended the framework of tripolar complex fuzzy sets (T CFS by in-
troducing the concept of tripolar complex fuzzy Lie brackets and investigating their funda-
mental algebraic properties. We demonstrated that the scalar multiplication and addition
of tripolar complex fuzzy Lie subalgebras yield a tripolar complex fuzzy Lie subalgebra,



M. Balamurugan, G. Ellammal, A. Iampan / Eur. J. Pure Appl. Math, 18 (3) (2025), 6489 24 of 26

reinforcing their algebraic closure properties. Also, we established that the homomor-
phic image of a nilpotent (or solvable) tripolar complex fuzzy Lie ideal remains nilpotent
(or solvable), highlighting the structural preservation under homomorphisms. Finally, we
proved that every nilpotent tripolar complex fuzzy Lie ideal is solvable, extending a fun-
damental result from classical Lie theory to the tripolar complex fuzzy setting. These
results contribute to the growing body of research on fuzzy algebraic structures, particu-
larly in the context of multipolar and complex-valued fuzzy systems. In future work, we
will extend the framework to n-polar complex Pythagorean fuzzy Lie algebras to explore
higher-dimensional algebraic properties.

Acknowledgements

This research was supported by University of Phayao and Thailand Science Research
and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567).

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