A Study of Normal Fuzzy Subgroups and Characteristic Fuzzy Subgroups of a Fuzzy Group

Ajmal, Naseem; Jahan, Iffat

doi:10.1007/s12543-012-0106-0

Cited by 17 publications

(15 citation statements)

References 15 publications

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“…Definition 5 (see [5]). Let A be a CFSG (G) and m ∈ G. en, the complex fuzzy left coset of A in G is represented by mAand is given by mA(g) � μ A (m −1 g): g ∈ G .…”

Section: Preliminariesmentioning

confidence: 99%

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On Structural Properties of $ξ$ -Complex Fuzzy Sets and Their Applications

et al. 2020

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Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ .

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“…Definition 5 (see [5]). Let A be a CFSG (G) and m ∈ G. en, the complex fuzzy left coset of A in G is represented by mAand is given by mA(g) � μ A (m −1 g): g ∈ G .…”

Section: Preliminariesmentioning

confidence: 99%

“…Mashour et al [4] studied many important features of normal fuzzy subgroups. For more details about the recent development of fuzzy subgroups, we refer to [5][6][7]. In addition, another important aspect of fuzzy sets is the level sets of this notion.…”

Section: Introductionmentioning

confidence: 99%

On Structural Properties of $ξ$ -Complex Fuzzy Sets and Their Applications

et al. 2020

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“…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%

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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“…This setting has its own limitations and does not even allow the formulation of various concepts of classical algebra in fuzzy or L-(lattice valued fuzzy) setting. This drawback can be removed easily if the parent structure considered in the definition of an L(fuzzy)-algebraic concept is an L(fuzzy)-algebraic structure rather than an ordinary algebraic structure [3]. In fact, very few researchers such as Martinez [12] have studied the properties of a L-subring of an L-ring.In [14,15], Ajmal and Prajapati have introduced the notion of maximal L-ideals of an L-ring with an essence similar to classical ring theory.…”

Section: Introductionmentioning

confidence: 99%

“…However, in the studies of fuzzy groups such an effort is lacking. Recently, a systematic study of L-subgroups (lattice valued fuzzy subgroups) of an L-group has been carried out in a series of papers [3,4,5,6,7] wherein a number of concepts of classical group theory have been extended to L-setting specially keeping in view their compatibility. The present paper is an endeavour to develop and study the maximal L-subgroup of an L-group along with its application to the notion of Frattini subgroups.…”

Section: Introductionmentioning

confidence: 99%

Maximal and Frattini L-Subgroups of an L-Group

Jahan¹,

Manas²

2020

Preprint

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In this paper, the concept of a maximal L-subgroup of an L-group has been defined in the spirit of classical group theory. Then, a level subset characterization has been established for the same. Then, this notion of maximal L-subgroups has been used to define Frattini L-subgroups. Further, the concept of non-generators of an L-group has been developed and its relation with the Frattini L-subgroup of an L-group has been established like their classical counterparts. Moreover, several properties pertaining to the concepts of maximal L-subgroups and Frattini L-subgroup have also been investigated. These two notions have been illustrated through several examples.

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A Study of Normal Fuzzy Subgroups and Characteristic Fuzzy Subgroups of a Fuzzy Group

Cited by 17 publications

References 15 publications

On Structural Properties of $ξ$ -Complex Fuzzy Sets and Their Applications

On Structural Properties of $ξ$ -Complex Fuzzy Sets and Their Applications

Development of L-Group Theory

Maximal and Frattini L-Subgroups of an L-Group

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A Study of Normal Fuzzy Subgroups and Characteristic Fuzzy Subgroups of a Fuzzy Group

Cited by 17 publications

References 15 publications

On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

Development of L-Group Theory

Maximal and Frattini L-Subgroups of an L-Group

Contact Info

Product

Resources

About

On Structural Properties of $ξ$ -Complex Fuzzy Sets and Their Applications

On Structural Properties of $ξ$ -Complex Fuzzy Sets and Their Applications