International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(1): 61-73
Published online March 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.1.61
© The Korean Institute of Intelligent Systems
Iffat Jahan^{1} and Ananya Manas^{2}
^{1}Department of Mathematics, Ramjas College, University of Delhi, Delhi, India
^{2}Department of Mathematics, University of Delhi, Delhi, India
Correspondence to :
Iffat Jahan (ij.umar@yahoo.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, the concept of pronormal lattice-valued fuzzy (L)-subgroups of an L-group was developed using the concept of conjugate L-subgroups developed in [11]. The concept of pronormal L-subgroups was investigated in the context of their normality and subnormality, and several related properties were established. Moreover, the relationship between pronormality with normalizers and maximal L-subgroups was explored.
Keywords: L-algebra, L-subgroup, Generated L-subgroup, Pronormal L-subgroup, Normal L-subgroup, Conjugate of L-subgroup, Maximal L-subgroup
The concept of a fuzzy set was developed by Zadeh [1] in 1965, and it was applied by Rosenfeld [2] to group theory in 1971, which led to the evolution of fuzzy group theory. In 1982, Liu [3] introduced the concept of lattice-valued fuzzy (
In classical group theory, pronormal subgroups are closely related to normal and subnormal subgroups. Pronormality together with subnormality is equivalent to normality. Pronormal fuzzy subgroups were introduced by Abou-Zaid [10] using level subsets. However, his definition and the corresponding study on pronormal fuzzy subgroups failed to provide any information on their deeper structures. Hence, a new approach for pronormal
Section 2 presents a preliminary discussion. Section 3 provides the definition of pronormal
Throughout this paper,
In 1965, Zadeh [1] introduced the concept of a fuzzy subset of a set. In 1967, Goguen [14] extended this concept to
Let
for each
For
For
Let
Note that if
Let
and
As previously, if
Throughout this paper,
In 1971, Rosenfeld [2] applied the concept of fuzzy sets to groups to introduce fuzzy subgroups of a group. In 1981, Liu [3] extended the concept of fuzzy subgroups to
If
(1)
(2)
The set of
If
The intersection of an arbitrary family of
If
Let
In the following,
Let
Let
If
In 1981, Wu [16] introduced normal fuzzy subgroups of a fuzzy group. To develop this concept, Wu [16] preferred the
If
The set of normal
Note that as previously, an arbitrary intersection of the family of normal
If
If
Finally, consider the following form [17]:
Let
Thus,
Let
The concept of pronormal subgroups in classical group theory utilizes the concept of conjugate subgroups. In fuzzy group theory, conjugate fuzzy subgroups were introduced by Mukherjee and Bhattacharya [18]. In their method, the conjugate of an
In [11], the authors introduced the conjugate of an
Let
Note that for an
Considering the level subset characterization for conjugate
If
In the following, the concept of pronormal
Let
Here, we note that for
Our definition of pronormal
If
(⇒) Let
First, because
If
Let
(⇐) If
Hence, there exists
where
Thus, we conclude that
In the following, we provide an example to demonstrate a pronormal
Let
We define an
Because
Because
Thus,
Hence,
Thus,
Therefore, ⟨
However,
Theorem 3.7 discusses the image of a pronormal
Let
Let
for all
Let
Let
Let
Hence,
First,
Let
This implies
where
where
Hence,
Hence,
In the following, we provide a level subset characterization for pronormal
Let
Let
First, because
Hence,
According to Theorem 2.10,
Let
i.e., (
As previously, because
that is,
This implies
Finally, we show that
Thus,
It follows that
For reverse inclusion, let
This implies
In this section, we present the various relationships between pronormal
In Theorem 4.2, we prove that a normal
Let
Let
Let
Let
If
Hence,
We conclude that
Considering
The normalizer of an
Theorem 4.6 discusses the pronormality of the normalizer of a pronormal
Let
The definition of the set product of two
and
Let
Reference [11] provides a new definition for the normalizer of an
Let
Let
Let
First, note that for all
Thus,
We show that
Therefore,
In Theorem 4.8, we show that the set product of a normal
Let
Let
Because
First, we show that (
Now,
We assume that
Hence,
We demonstrate that (
Thus,
Moreover, because
Hence, from Lemma 4.1, (
Here, noticeably
Thus,
Thus, we conclude that
In the following, we show that every maximal
We assume that
If
Let
By maximality of
Hence, in both cases,
In Theorem 4.15, we present the main result of this study, i.e., an
If
which is the conjugate of
Moreover,
Let
Thus,
is the normal closure series of
Clearly,
Here, we prove the following:
Let
If
Because
The following result is obtained from the definition of pronormal
Let
Let
Let
Suppose that
be the normal closure series for
Because
that is,
Because
Therefore,
Suppose that the result holds for
Suppose that
as a normal closure series for
represents the normal closure series for
An important application of Theorem 4.15 is in nilpotent
The concept of a nilpotent
Let
The commutator
Let
of
Let
terminates with trivial
Here, we define the successive normalizers of
According to the definition of a normalizer (see Definition 4.4),
is an ascending chain of
Let
Let
terminating at
Based on Lemmas 4.19 and 4.20, we have the following:
Let
Thus, Theorems 4.2, 4.15, and 4.21 yield the following result:
Let
According to classical group theory, pronormal subgroups play an indispensable role in studying normality and subnormality. Although the concepts of normal and subnormal
Research in fuzzy group theory stopped after Tom Head’s metatheorem and subdirect product theorems. This is because most concepts and results of fuzzy algebra can be established by simple applications of the metatheorem and subdirect product theorem. However, the metatheorem and subdirect product theorems are not applicable to the
No potential conflict of interest relevant to this article was reported.
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(1): 61-73
Published online March 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.1.61
Copyright © The Korean Institute of Intelligent Systems.
Iffat Jahan^{1} and Ananya Manas^{2}
^{1}Department of Mathematics, Ramjas College, University of Delhi, Delhi, India
^{2}Department of Mathematics, University of Delhi, Delhi, India
Correspondence to:Iffat Jahan (ij.umar@yahoo.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, the concept of pronormal lattice-valued fuzzy (L)-subgroups of an L-group was developed using the concept of conjugate L-subgroups developed in [11]. The concept of pronormal L-subgroups was investigated in the context of their normality and subnormality, and several related properties were established. Moreover, the relationship between pronormality with normalizers and maximal L-subgroups was explored.
Keywords: L-algebra, L-subgroup, Generated L-subgroup, Pronormal L-subgroup, Normal L-subgroup, Conjugate of L-subgroup, Maximal L-subgroup
The concept of a fuzzy set was developed by Zadeh [1] in 1965, and it was applied by Rosenfeld [2] to group theory in 1971, which led to the evolution of fuzzy group theory. In 1982, Liu [3] introduced the concept of lattice-valued fuzzy (
In classical group theory, pronormal subgroups are closely related to normal and subnormal subgroups. Pronormality together with subnormality is equivalent to normality. Pronormal fuzzy subgroups were introduced by Abou-Zaid [10] using level subsets. However, his definition and the corresponding study on pronormal fuzzy subgroups failed to provide any information on their deeper structures. Hence, a new approach for pronormal
Section 2 presents a preliminary discussion. Section 3 provides the definition of pronormal
Throughout this paper,
In 1965, Zadeh [1] introduced the concept of a fuzzy subset of a set. In 1967, Goguen [14] extended this concept to
Let
for each
For
For
Let
Note that if
Let
and
As previously, if
Throughout this paper,
In 1971, Rosenfeld [2] applied the concept of fuzzy sets to groups to introduce fuzzy subgroups of a group. In 1981, Liu [3] extended the concept of fuzzy subgroups to
If
(1)
(2)
The set of
If
The intersection of an arbitrary family of
If
Let
In the following,
Let
Let
If
In 1981, Wu [16] introduced normal fuzzy subgroups of a fuzzy group. To develop this concept, Wu [16] preferred the
If
The set of normal
Note that as previously, an arbitrary intersection of the family of normal
If
If
Finally, consider the following form [17]:
Let
Thus,
Let
The concept of pronormal subgroups in classical group theory utilizes the concept of conjugate subgroups. In fuzzy group theory, conjugate fuzzy subgroups were introduced by Mukherjee and Bhattacharya [18]. In their method, the conjugate of an
In [11], the authors introduced the conjugate of an
Let
Note that for an
Considering the level subset characterization for conjugate
If
In the following, the concept of pronormal
Let
Here, we note that for
Our definition of pronormal
If
(⇒) Let
First, because
If
Let
(⇐) If
Hence, there exists
where
Thus, we conclude that
In the following, we provide an example to demonstrate a pronormal
Let
We define an
Because
Because
Thus,
Hence,
Thus,
Therefore, ⟨
However,
Theorem 3.7 discusses the image of a pronormal
Let
Let
for all
Let
Let
Let
Hence,
First,
Let
This implies
where
where
Hence,
Hence,
In the following, we provide a level subset characterization for pronormal
Let
Let
First, because
Hence,
According to Theorem 2.10,
Let
i.e., (
As previously, because
that is,
This implies
Finally, we show that
Thus,
It follows that
For reverse inclusion, let
This implies
In this section, we present the various relationships between pronormal
In Theorem 4.2, we prove that a normal
Let
Let
Let
Let
If
Hence,
We conclude that
Considering
The normalizer of an
Theorem 4.6 discusses the pronormality of the normalizer of a pronormal
Let
The definition of the set product of two
and
Let
Reference [11] provides a new definition for the normalizer of an
Let
Let
Let
First, note that for all
Thus,
We show that
Therefore,
In Theorem 4.8, we show that the set product of a normal
Let
Let
Because
First, we show that (
Now,
We assume that
Hence,
We demonstrate that (
Thus,
Moreover, because
Hence, from Lemma 4.1, (
Here, noticeably
Thus,
Thus, we conclude that
In the following, we show that every maximal
We assume that
If
Let
By maximality of
Hence, in both cases,
In Theorem 4.15, we present the main result of this study, i.e., an
If
which is the conjugate of
Moreover,
Let
Thus,
is the normal closure series of
Clearly,
Here, we prove the following:
Let
If
Because
The following result is obtained from the definition of pronormal
Let
Let
Let
Suppose that
be the normal closure series for
Because
that is,
Because
Therefore,
Suppose that the result holds for
Suppose that
as a normal closure series for
represents the normal closure series for
An important application of Theorem 4.15 is in nilpotent
The concept of a nilpotent
Let
The commutator
Let
of
Let
terminates with trivial
Here, we define the successive normalizers of
According to the definition of a normalizer (see Definition 4.4),
is an ascending chain of
Let
Let
terminating at
Based on Lemmas 4.19 and 4.20, we have the following:
Let
Thus, Theorems 4.2, 4.15, and 4.21 yield the following result:
Let
According to classical group theory, pronormal subgroups play an indispensable role in studying normality and subnormality. Although the concepts of normal and subnormal
Research in fuzzy group theory stopped after Tom Head’s metatheorem and subdirect product theorems. This is because most concepts and results of fuzzy algebra can be established by simple applications of the metatheorem and subdirect product theorem. However, the metatheorem and subdirect product theorems are not applicable to the
The lattice of
The lattice of