Normal fuzzy subgroups and fuzzy normal series of finite groups Fuzzy sets and systems 72 (1995) 379-383

1997 ◽  
Vol 86 (1) ◽  
pp. 121
Author(s):  
M.Atif A. Mishref
Keyword(s):  
Fuzzy Sets ◽  
Finite Groups ◽  
Normal Series ◽  
Fuzzy Subgroups

Image development in the framework of ξ –complex fuzzy morphisms

2021 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Unit Disk ◽  
Homomorphic Image ◽  
Significant Role ◽  
Original Form ◽  
Mathematical Tool ◽  
Fuzzy Subgroups ◽  
Fundamental Theorems

The study of complex fuzzy sets defined over the meet operator (ξ –CFS) is a useful mathematical tool in which range of degrees is extended from [0, 1] to complex plane with unit disk. These particular complex fuzzy sets plays a significant role in solving various decision making problems as these particular sets are powerful extensions of classical fuzzy sets. In this paper, we define ξ –CFS and propose the notion of complex fuzzy subgroups defined over ξ –CFS (ξ –CFSG) along with their various fundamental algebraic characteristics. We extend the study of this idea by defining the concepts of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –complex fuzzy subgroups and establish fundamental theorems of ξ –complex fuzzy morphisms. In addition, we effectively apply the idea of ξ –complex fuzzy homomorphism to refine the corrupted homomorphic image by eliminating its distortions in order to obtain its original form. Moreover, to view the true advantage of ξ –complex fuzzy homomorphism, we present a comparative analysis with the existing knowledge of complex fuzzy homomorphism which enables us to choose this particular approach to solve many decision-making problems.

scholarly journals Counting fuzzy subgroups of some finite groups by a new equivalence relation

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6151-6160
Author(s):  
Ardekani Kamali
Keyword(s):  
Group Theory ◽  
Finite Groups ◽  
Equivalence Relation ◽  
Theoretical Foundation ◽  
Dihedral Groups ◽  
Significant Aspect ◽  
Exact Number ◽  
Consistent Group ◽  
Fuzzy Subgroups

The study concerning the classification of the fuzzy subgroups of finite groups is a significant aspect of fuzzy group theory. In early papers, the number of distinct fuzzy subgroups of some nonabelian groups is calculated by the natural equivalence relation. In this paper, we treat to classifying fuzzy subgroups of some groups by a new equivalence relation which has a consistent group theoretical foundation. In fact, we determine exact number of fuzzy subgroups of finite non-abelian groups of order p3 and special classes of dihedral groups.

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

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Hanan Alolaiyan ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Quotient Group ◽  
Physical Situation ◽  
The Novel ◽  
Complex Fuzzy Set ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
Necessary And Sufficient

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 .

Gradewise Properties of Subgroups and Their Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 255-266
Author(s):  
Wenbin Guo ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Binary Relation ◽  
Finite Groups ◽  
Normal Series ◽  
A Finite Group

For each finite group E, let Θ(E) be a binary relation on the set of all subgroups of E. If A and B are subgroups of a finite group G, then we say that the pair (A, B) enjoys the gradewise property (resp., generalized gradewise property) Θ in G if G has a normal series Γ : 1 = G0 [Formula: see text] such that for each [Formula: see text], we have [Formula: see text] (resp., we have [Formula: see text]. Using these concepts, we obtain some new characterizations for solubility and supersolubility of finite groups and generalize some known results.

scholarly journals Pinned-flags of some operations on fuzzy subgroups

2005 ◽  
Vol 2005 (23) ◽  
pp. 3819-3826
Author(s):  
B. B. Makamba ◽  
V. Murali
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Fuzzy Subgroups

Fuzzy subgroups of finite groups have been treated recently using the concept of pinned-flags. In this paper, we consider the operations of intersection, sum, product, and quotient of fuzzy subgroups of finite abelian groups in general, in terms of pinned-flags. We develop algorithms to construct pinned-flags of fuzzy subgroups corresponding to these operations and prove their validity. We illustrate some applications of such algorithms.

scholarly journals A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 992
Author(s):  
Hanan Alolaiyan ◽  
Halimah A. Alshehri ◽  
Muhammad Haris Mateen ◽  
Dragan Pamucar ◽  
Muhammad Gulzar
Keyword(s):  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Classical Group ◽  
Machine Learning Algorithms ◽  
Normal Subgroups ◽  
Specific Complex ◽  
Alternative Conceptualization ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
Bipolar Fuzzy Sets

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (α,β)-complex fuzzy sets and then define α,β-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of α,β-complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

Some notes on equivalent fuzzy sets and fuzzy subgroups

2005 ◽  
Vol 152 (2) ◽  
pp. 403-409 ◽  
Author(s):  
Chen Degang ◽  
Jiang Jiashang ◽  
Wu Congxin ◽  
E.C.C. Tsang
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Subgroups
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