Some Structure Properties of the Cyclic Fuzzy Group Family

2005 ◽  
Vol 12 (03) ◽  
pp. 471-476
Author(s):  
Hacı Aktaş ◽  
Naim Çağman
Keyword(s):  
Fuzzy Sets ◽  
Cyclic Group ◽  
Fuzzy Group ◽  
Classical Notion ◽  
Fuzzy Subgroups ◽  
Group Family ◽  
Structure Properties

In crisp environment, the notion of cyclic group on a set is well known. We study an extension of this classical notion to the fuzzy sets to define the concept of cyclic fuzzy subgroups. By using these cyclic fuzzy subgroups, we then define a cyclic fuzzy group family and investigate its structure properties.

scholarly journals Complex Fuzzy Groups Based on Rosenfeld’s Approach

2021 ◽  
Vol 20 ◽  
pp. 368-377
Author(s):  
Eman A. Abuhijleh ◽  
Mourad Massa’deh ◽  
Amani Sheimat ◽  
Abdulazeez Alkouri
Keyword(s):  
Group Theory ◽  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Unit Disk ◽  
Mathematical Tool ◽  
Membership Functions ◽  
Fuzzy Group ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups

Complex fuzzy sets (CFS) generalize traditional fuzzy sets (FS) since the membership functions of CFS reduces to the membership functions of FS. FS values are always at [0, 1], unlike CFS which has values in the unit disk of C. This paper merges notion and concept in group theory and presents the notion of a complex fuzzy subgroup of a group. This proposed idea represents a more general and better optional mathematical tool as one of the approaches in the fuzzy group. However, this research defines the notion of complex fuzzy subgroupiod, complex fuzzy normal subgroup, and complex fuzzy left(right) ideal. Therefore, the lattice, homomorphic preimage, and image of complex fuzzy subgroupiod and ideal are introduced and studied its properties. Finally, complex fuzzy subgroups and their properties are presented and investigated

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 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 .

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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