Subnormality and Theory of L-subgroups

Jahan, Iffat; Ajmal, Naseem; Davvaz, Bijan

doi:10.29020/nybg.ejpam.v15i4.4548

Cited by 4 publications

(7 citation statements)

References 4 publications

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“…y, z ∈ G η y ðÞ ∧μ z ðÞ fg for each x ∈ G: Before we embark on the study of normal closure series and subnormality in Lsetting, we generalize notion of conjugacy and provide the definition of a conjugate of an L-subset by an L-subset (see [10,33]…”

Section: Normal Closure Series and Subnormalitymentioning

confidence: 99%

“…Then, this characterization is used to establish that when the lattice L is an upper well ordered chain, then every L-subgroup of a nilpotent L-group is subnormal. For this purpose, we need to develop a necessary mechanism (see [10,33]). Here, we present:…”

Section: Subnormal L-subgroups and Nilpotencymentioning

confidence: 99%

“…Afterwards, this idea was further taken up only by Martinez [23,24]. Following this approach, the theory of L-subgroups is developed in a systematic and consistent manner in our papers [10,[25][26][27][28][29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

“…For the sake of completeness, few definitions from lattice theory have also been incorporated. In papers [10,[25][26][27][28][29][30][31][32][33][34], various concepts of L-group theory have been explored.…”

Section: Introductionmentioning

confidence: 99%

“…Here, it has been shown that the normal closure series, defined in this work, is the fastest descending normal series containing given L-subgroup. This sets the ground for the development of subnormality in L-group theory [10,33]. During the course of the development of subnormality, it has also been proved that every L-subgroup of a nilpotent L-subgroup is a subnormal L-subgroup.…”

Section: Introductionmentioning

confidence: 99%

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Development of L-Group Theory

Jahan¹

2023

Advances in Fuzzy Logic Systems

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In this work, we present a systematic and successful development of L-group theory. A universal construction of a generated L-subgroup has been provided by using level subsets of given L-subsets. This construction allows us to define and study commutator L-subgroups, normalizer of an L-subgroup, nilpotent L-subgroups, solvable L-subgroups, normal closure of an L-subgroup. All these concepts and their inter-relationships have been presented. Here we mention that in this work we also exhibit a characterization of solvable L-subgroup with the help of a series of L-subgroups such that at each level, the factor groups of level subgroups of their consecutive members are Abelian. This allows us to introduce the notion of a supersolvable L-subgroup by using the factors of level subgroups at each level of a subinvariant series of an L-subgroup. Also, by using successive normal closures, we transfinitely define a series called the normal closure series of the L-subgroup. It has been shown that it is the fastest descending normal series containing given L-subgroup. This sets the ground for the development of subnormality in L-group theory. In the last, we study the notion of subnormal L-subgroups.

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Section: Normal Closure Series and Subnormalitymentioning

confidence: 99%

Section: Subnormal L-subgroups and Nilpotencymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Development of L-Group Theory

Jahan¹

2023

Advances in Fuzzy Logic Systems

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The join problem of subnormal L -subgroups of an L -group

Jahan,

Ajmal

2023

Research in Mathematics

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Associated Prime Ideal and Minimal Prime Ideal of an Ideal of an L-Subring

Prajapati,

Ajmal,

Jahan

2023

Fuzzy Information and Engineering

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In this paper, a systematic theory for the ideals of an L-ring has been developed. Earlier the authors have introduced the concepts of prime ideals, semiprime ideals, primary ideals, and radical of an ideal in an Lring. They have also introduced and discussed the notion of a maximal ideal in different papers wherein several results pertaining to these notions have been proved. In this paper, the concepts of associated prime ideal, minimal prime ideal, and that of irreducibility of an ideal in an L-ring have been introduced, which in fact, are the continuation of authors' previous works in the theory of ideals in L-rings. KEYWORDSprime ideal; semiprime ideal; primary ideal; radical of an ideal; L-subring S o far in the studies of fuzzy algebraic substructures, the definition of a subalgebra has been formulated in the framework of classical algebra. In fact, from Rosenfeld [1] onwards, almost all the researchers defined and studied the notion of a fuzzy subgroup of an ordinary group. Similar is the situation in the theory of fuzzy rings or lattice valued fuzzy rings [2−6] . Here we mention that such studies were initiated by Liu [7] . In Refs. [8,9] and in the theory of L-subgroups [10−17] , there is a deviation from this approach. Wu [18] gave a hint of pursuing studies of a fuzzy subgroup in a fuzzy group by providing the definition of a fuzzy normal subgroup in a fuzzy group. This idea was further taken up by Martinez [19,20] . In fact, in Ref.[10], we studied the notion of a normal fuzzy subgroup of a fuzzy group and defined a characteristic fuzzy subgroup of a fuzzy group. This laid the foundation of the theory of fuzzy subalgebras where the parent structures are also fuzzy algebras. The development of fuzzy algebraic structures was hampered mainly because of two reasons. Firstly, it was due to some inherent problem that the analogs of some of the concepts and results of classical algebra were not even formulated in this setting such as transitivity of properties of subsets of an algebra. For example: "a normal subgroup of a characteristic subgroup of a group is a normal subgroup of the given group"; and "a prime ideal of an ideal of a ring is an ideal of the given ring". Moreover, the concept of maximality of a fuzzy substructure of an algebraic structure is not properly formulated. Secondly, since most of the concepts studied in fuzzy algebraic

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Subnormality and Theory of L-subgroups

Cited by 4 publications

References 4 publications

Development of L-Group Theory

Development of L-Group Theory

The join problem of subnormal L -subgroups of an L -group

Associated Prime Ideal and Minimal Prime Ideal of an Ideal of an L-Subring

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