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Q-Fuzzy Implementations of N- Lie Ideals Structures Over Lie
Subalgebas -A New Approach
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.2026.150600090
Received: 10 June 2026; Accepted: 15 June 2026; Published: 08 July 2026
ABSTRACT
In this paper, we study the concept of q-fuzzy n-lie sub algebras (ideals) of lie algebras and investigate some of
their properties. We discuss the relationship between q-fuzzy lie sub algebra (respectively ideals) and Lie sub
algebras (respectively ideals). For a finite number of q-fuzzy n-lie sub algebras, we construct new q-fuzzy n-lie
sub algebras on their direct sum. Finally, we analyze the homomorphism concept of q-fuzzy n-lie sub algebra
over ideals.
Key words: Q-fuzzy set, sub algebra, q-fuzzy n-ideal, direct sum, homomorphism, Lie algebra, vector space.
AMS subject classification (2000): 04A72, 17B99.
INTRODUCTION
The inception of the concept of Lie algebras was performed by one of the renowned Norwegian mathematicians
of the 19th century, Sophus Lie (1842–1899) [3,4]. After him, the idea has undergone many developments,
modifications, and additions, by multiple authors who incorporated further advanced and novel types and factors
with this concept, one of which is the notion of Hom–Lie algebras. The base of the idea of Hom–Lie algebras
was initially formulated by Hartwig, Larsson, and Silvestrov in 2006 [5]. It lies among the generalities of the
notion of classical Lie algebras. In recent years, Hom–Lie algebras have turned into a stimulating area of physics
and mathematics. For further deliberations concerning Hom–Lie algebras, we refer the reader to [5,6 ,7 ,8 ,9 ,10
,11].In terms of the case of fuzzy sets, Zadeh [12] was the first one to present this notion. The application of
fuzzy set theory extends to various fields including decision theory [13,14], logic [15], soil science [16],
computer science [17], social life [18,19], artificial intelligence [20,21 ,22] , management, various disciplines.
Originally, the study of fuzzy Lie sub algebras of Lie algebras was first introduced by Yehia [25] in the year
1996. Afterwards, numerous authors [26,27 ,28 ,29 ,30 ,31 ,32 ,33 ,34], as well as the references within) have
made contributions to the concept of fuzzy sets (in more general, complex fuzzy sets and intuitionistic fuzzy
sets) by applying them in a variety of directions in Lie algebras. Lie algebras was also discovered by him when
he attempted to classify certain smooth subgroups of a general linear group. The importance of lie algebras in
mathematics and physics has become increasingly evident in recent years. In applied mathematics, Lie theory
remains a powerful tool for studying differential equations, special functions and perturbation theory. It is noted
that lie theory has application not only in mathematics and physics but also in divergent fields such as continuum
mechanics, cosmology and life sciences. Lie algebra has nowadays even been applied by electrical engineers in
solving problems in mobile robot control. On the other hand, Zadeh [12.15] introduce the notion of fuzzy subset
of a set in 1965. By using fuzzy sets, people have established the theory for study uncertainty. The study of fuzzy
Lie algebras was initiated by Yehia [25] in 1996. In this paper, we study the concept of q-fuzzy n-lie sub
algebras(ideals) of lie algebras and investigate some of their properties. We discuss the relationship between q-
fuzzy lie sub algebra(respectively ideals) and Lie sub algebras (respectively ideals). For a finite number of q-
fuzzy n-lie sub algebras, we construct a new q-fuzzy n-lie sub algebras on thir direct sum. Finally, we analyse
the homomorphism concept of q-fuzzy n-lie sub algebra over ideals.
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Preliminaries
Definition-2.1: Fuzzy set: Let be a non-empty set. A fuzzy set drawn from is defined as
, where
is the member function of the fuzzy set .
Definition-2.2: If is a fuzzy subset of , for
, then the set
is called level subset of of with respect to a fuzzy subset .
Definition-2.3: A fuzzy set
is a non-empty fuzzy subset, if is not constant function.
Definition-2.4 (Lie algebra) : Let F be a field. A Lie algebra over F is triple,
where is a vector space
over F, is a linear map, and
is a linear map, satisfying the following properties.
(i)
for all (Skew- symmetry property)
(ii)
+
(Jacobi identity).
It is clear that every Lie algebra is a q-fuzzy Lie algebra by setting .
Example 2.5: Let be a vector space over F and
be any Skew- symmetric bilinear map. If
is the zero map, then
is a Lie algebra.
Definition-2.6: A linear map
is called a morphismof Lie algebra if the following two identities are
satisfied.
(i)
, for all
(ii)
Note: Throughout this paper is a Lie algebra over F.
Let
. For the sake of simplicity we use the symbols and to denote
and
respectively.
Definition-2.7: A Q-fuzzy set ‘’ on is a q-fuzzy n-Lie subalgebra if the following conditions are satisfied for
all and is a linear map;
(i)
,
(ii)
,
(iii)
.
If the condition (iii) is replaced by
, then ‘’ is called a q-fuzzy n-Lie
ideal of . It is clear that if ‘’ is a q-fuzzy n-Lie ideal of , then it is a q-fuzzy n-Lie subalgebral of .
Example-2.8: Let be a vector space with basis
. We define the twisted map
by setting
and
. Let
be any Skew- symmetric bilinear
map such that
,
, and
for all states =1,2,3, then
is a q-fuzzy n-Lie algebra.
Indeed, for each , we have
is a scalar multiple of
. Also
is scalar multiple of
for
each . Therefore
for each . This implies that q-fuzzy n-Jacobi identity is
satisfied.
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We define ‘’ as follows
Then ‘’ is a q-fuzzy n-Lie ideal of .
Let V be a vector space and ‘’ be a Q-fuzzy set on it. For
, the set
is called upper level of . The following theorem will show a relation
between q-fuzzy n-Lie sub algebra of and Lie sub algebra of .
3. Some Results on Q-FUZZY N-LIE Subalgebras
Theorem-3.1: Let ‘’ be a Q-fuzzy subset of . Then the following statements are equivalent;
(i) ‘’ is a q-fuzzy n-Lie subalgebra of .
(ii) The non-empty set
is a q-fuzzy n-Lie sub algebra of for every
. Proof:Let
,
and let
.
As ‘’ is a q-fuzzy n-Lie sub algebra of , we have
,
,
,
and so , and
are elements in
.
Conversely, let
be a q-fuzzy n-Lie sub algebra of , for every
.
Let . We may assume that
. So
.
As
is a sub algebra of , we have is in
, and
.
Since
is a q-fuzzy n-Lie sub algebra of , we have
and are in
.
Hence,
and
.
Theorem-3.2: Let ‘’ be a Q-fuzzy subset of . Then the following statements are equivalent;
(i) ‘’ is a q-fuzzy n-Lie ideal of .
(ii) The non-empty set
is a q-fuzzy n-Lie ideal of for every
Proof: Let ‘’ is a q-fuzzy n-Lie ideal of .
Then it is a q-fuzzy n-Lie subalgebra of .
According to the theorem-3.1, every
, we have , are in
.
For and
, we find
.
That is
.
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Conversely, assume that every
is a Q-fuzzy Lie ideal of , then
is a Q-fuzzy Lie algebra of . Thus, we can proceed as in the theorem above and the only difference appears in
the proof of the following statement;
, for all .
Let . Without loss of generality, we may assume that
.
Set
. Hence
.
As
is a q-fuzzy Lie ideal of , we have
.
This shows that
.
Let V be a vector space. For
and a Q-fuzzy set on V, the set
is called strong upper level of . We have following result.
Theorem-3.3: Let ‘’ be a Q-fuzzy subset of . Then the following statements are equivalent;
(i) ‘’ is a q-fuzzy n-Lie subspace of .
(ii) The strong upper level
is a sub algebra of for every
.
Proof: For
, let
.
As ‘’ is a q-fuzzy n-Lie subspace of , we have
,
,
.
Consequently, , and
are elements in
.
Conversely, assume that for every
, we have
is a Lie-subalgebra of . Let .
We need to show that the conditions of definition ** are satisfied .
If
or
, then
.
Suppose that
and
.
Suppose to the contrary that
and
are less than the value of
. Let
be the greatest lower bound of the set
and less than the value of
.
We have ,
, and hence
.
This contradicts that there are no element with
.
This shows that
Again let ‘
’ be the largest number of
such that
and there are no with
. As
is q-fuzzy n-Lie subspace, we have and
and so
. Thus
is greater than or equal to
.
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Using almost the same argument one can show the following result.
Construction of Direct Sum of Q-FUZZY N-LIE Algebras
Given q-fuzzy m-Lie algebras
, =1,2,…,m, then
is a q-fuzzy Lie algebra by setting
,
and the linear map
In the special case when , we obtain
.
Let
,
, …….,
be q-fuzzy Lie algebras.
Suppose that
,
,…,
are Q-fuzzy subsets of
,
,…,
respectively. Then the generalized
Cartesian sum of Q-fuzzy sets induced by
,
,…,
on
is
;
.
Theorem-4.1: Let
,
, …….,
be q-fuzzy Lie algebras. Let
,
,…,
be
Q-fuzzy n-Lie sub algebras of
,
,…,
respectively, then
is Q-fuzzy n-Lie sub algebras of
.
Proof:Let
,
.
.
Also,
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.
The rest of the proof is similar. so we omit it.
However the direct sum of q-fuzzy n-Lie ideals of q-fuzzy Lie sub algebras
and
is not necessary to
be a q-fuzzy n-Lie ideal of the q-fuzzy Lie aubalgebra
.
5. Q-FUZZY N-LIE Algebras and Q-FUZZY N-LIE Algebras Morphisms
Suppose is a function. If
is a Q-fuzzy set on , then we can define Q-fuzzy set on
induced by and “
” by setting
for only . Also if
is a Q-fuzzy set on , then
is actually set on induced by and
in the setting of Lie algebras.
We extend it to q-fuzzy Lie algebra case.
Theorem-5.1: Let
be a morphism of q-fuzzy Lie algebras. If
is a q-fuzzy n-lie sub algeba (respectively ideal) of
, then the Q-fuzzy set
is also a q-fuzzy n-lie sub algeba (respectively ideal) of
.
Proof:Let
. Then
(Since is linear)
(Since
is a q-fuzzy n-lie sub algebra)
and
(Since is homomorphism)
(Since
is a q-fuzzy n-lie sub algebra)
.
Let
. Then
(Since is homomorphism)
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(Since
is a q-fuzzy n-lie sub algebra)
The case of q-fuzzy n-lie ideal is similar to show.
If
is a Lie algebra homomorphism and
is a Q-fuzzy sub algebra of
, then the image of ,
is a Q-fuzzy subalgebra of
. In the following theorem we consider an analogue result for the
case of q-fuzzy Lie algebras.
Theorem-5.2: Let
be a morphism from
to
. If
is a q-fuzzy n-Lie sub
algebras of
, then
is also a q-fuzzy n-Lie subalgebras of
.
Proof:Let
. As is onto, there are
such that
,
.
We get,
, and
.
Now, we find
.
Also,
.
.
Also,
.
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Chung-Gook Kim and Dong-Soo Lee
proved if is a surjective Lie algebra homomorphism and
If
is a fuzzy ideal of , then
is a fuzzy ideal of .
We will extend the result to q-fuzzy n-Lie subalgebra case.
Theorem-5.3: Let
be an ontomorphism of q-fuzzy Lie algebras. If
is a q-
fuzzy n-Lie ideal of
, then
is also a q-fuzzy n-Lie ideal of
.
Proof: The proof is similar to the proof of theorem above.
We only need to claim that
,
.
Let
and assume by contradiction, that
Then
or
.
We may assume, without loss of generality, that
.
Choose a number
such that
.
There is
with
.
As is onto, there exists
, we note that
.
Thus
>
which is contradiction.
CONCLUSION
We discussed q-fuzzy n-Lie sub algebra over ideal structures in this article. We constructed direct sum of q-
fuzzy n-ideals and homomorphism of q-fuzzy lie sub algebra is analysed.
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