Free Fuzzy Subgroups and Fuzzy Subgroup Presentations

2005 ◽  
pp. 119-138
Author(s):  
John N. Mordeson ◽  
Kiran R. Bhutani ◽  
Azriel Rosenfeld
Keyword(s):  
Fuzzy Subgroup ◽  
Fuzzy Subgroups

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 ANTI SUBGRUP α-FUZZY

2020 ◽  
Vol 14 (1) ◽  
pp. 10
Author(s):  
Fiqriani Noor ◽  
Saman Abdurrahman ◽  
Naimah Hijriati
Keyword(s):  
Normal Subgroup ◽  
Group Structure ◽  
Normal Subgroups ◽  
Fuzzy Subset ◽  
New Concepts ◽  
Fuzzy Algebra ◽  
Sufficient And Necessary Conditions ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
The Relationship

The concept of fuzzy subgroups is a combination of the group structure with the fuzzy set, which was first introduced by Rosenfeld (1971). This concept became the basic concept in other the fuzzy algebra fields such as fuzzy normal subgroups, anti fuzzy subgroups and anti fuzzy normal subgroups. The development in the area of fuzzy algebra is characterized by the continual emergence of new concepts, one of which is the α-anti fuzzy subgroup concept. The idea of α-anti fuzzy subgroups is a combination between the α-anti fuzzy subset and anti fuzzy subgroups. The α-anti subset fuzzy which is an anti fuzzy subgroup is called as α-anti fuzzy subgroup. The purpose of this study is to prove that the α-anti fuzzy subset is an anti fuzzy subgroup, examine the relationship between α-anti fuzzy subgroups with anti fuzzy subgroups and α-fuzzy normal subgroups with anti fuzzy subgroups. The results of this study are, if A is an anti fuzzy subgroup (an anti fuzzy normal subgroup), then an α-anti subset fuzzy of A is an anti fuzzy subgroup (an anti fuzzy normal subgroup). However, this does not apply otherwise. Furthermore, this study also provides sufficient and necessary conditions for an α-anti fuzzy subset of any group to be an α-anti fuzzy subgroup and the formation of a group of factors that are built from an α-anti fuzzy normal subgroup.Keywords : Anti Fuzzy Subgroup, Anti Fuzzy Normal Subgroup, α-Anti Fuzzy Subgroup and α-Anti Fuzzy Normal Subgroup.

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.

scholarly journals Algebraic Properties of Implication-Based Anti-Fuzzy Subgroup over a Finite Group

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1116-1120
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Algebraic Properties ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
A Finite Group ◽  
Fuzzy Direct Product

This article deals with few algebraic characteristics of implication-based anti-fuzzy subgroup of a finite group.In addition, the implication-based anti-fuzzy direct product of implication-based anti-fuzzy subgroups over finite groups is developed and studied elaborately. The condition for an implication-based anti-fuzzy subgroup of a finite group to be a conjugate to another implication-based anti-fuzzy subgroup is conceptualized. Some of their characteristics are investigated in this paper.

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

Product of Implication-Based Intuitionistic Fuzzy Semiautomatons over Finite Groups

2019 ◽  
Vol 15 (03) ◽  
pp. 503-515 ◽  
Author(s):  
M. Selvarathi
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Direct Product ◽  
Finite Groups ◽  
Fundamental Properties ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
A Finite Group

In this paper, Implication-based intuitionistic fuzzy semiautomaton (IB-IFSA) of a finite group is defined and investigated. The theory of an implication-based intuitionistic fuzzy kernel and implication-based intuitionistic fuzzy subsemiautomaton of an IB-IFSA over a finite group are formulated using the approach of implication-based intuitionistic fuzzy subgroup and implication-based intuitionistic fuzzy normal subgroup. The product of implication-based intuitionistic fuzzy subgroups is postulated and investigated. Further, direct product of implication-based intuitionistic fuzzy semiautomatons over the finite groups is elaborately studied. Fundamental properties concerning them are also dealt with.

Introduction to Plithogenic Subgroup

2020 ◽  
pp. 213-259 ◽  
Author(s):  
Sudipta Gayen ◽  
Florentin Smarandache ◽  
Sripati Jha ◽  
Manoranjan Kumar Singh ◽  
Said Broumi ◽  
...  
Keyword(s):  
Algebraic Structures ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups

This chapter gives some essential scopes to study some plithogenic algebraic structures. Here the notion of plithogenic subgroup has been introduced and explored. It has been shown that subgroups defined earlier in the crisp, fuzzy, intuitionistic fuzzy, as well as neutrosophic environments, can also be represented as plithogenic fuzzy subgroups. Furthermore, introducing function in plithogenic setting, some homomorphic characteristics of plithogenic fuzzy subgroup have been studied. Also, the notions of plithogenic intuitionistic fuzzy subgroup, plithogenic neutrosophic subgroup have been introduced and their homomorphic characteristics have been analyzed.

scholarly journals Subgrup Normal Multi-Fuzzy, Subgrup Normal Multi-Anti Fuzzy, Dan Teorema Korespondensi

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Amaluddin Amaluddin ◽  
Budi Surodjo
Keyword(s):  
One To One ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups

In any group $ G $, we can form normal multi-fuzzy subgroups and normal multi-anti fuzzy subgroups. In this paper, we discuss the properties of normal multi-fuzzy subgroups and normal multi-anti fuzzy subgroups. Based on this structure, we construct the factor groups relative to a normal multi-fuzzy subgroup or normal multi-anti fuzzy subgroup. This gave rise to a one-to-one correspondence between the normal multi-anti fuzzy subgroups of a group $ G_2 $ and those of a group $ G_1 $ which are constant on the kernel of homomorphism $ f $. This correspondence can occur if $ f $ is epimorphism of groups from $ G_1 $ to $ G_2 $.

scholarly journals Notes on Bipolar-Valued Fuzzy Subgroups of a Group

2013 ◽  
Vol 7 ◽  
pp. 40-45
Author(s):  
M.S. Anitha ◽  
K.L. Muruganantha Prasad ◽  
K. Arjunan
Keyword(s):  
Fuzzy Subgroup ◽  
Fuzzy Subgroups

In this paper, we study some of the properties of bipolar-valued fuzzy subgroup and prove some results on these.

scholarly journals Anti Q-fuzzy Subgroups under t-conorms

2020 ◽  
pp. 13-28
Author(s):  
Rasul Rasuli
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups

In this paper we introduce the notion of anti Q-fuzzy subgroups of G with respect to t‑conorm C and study their important properties. Next we define the union, normal and direct product of two anti Q-fuzzy subgroups of G with respect to t-conorm C and we show that the union, normal and direct product of them is again an anti Q-fuzzy subgroup of G with respect to t-conorm C. It is also shown that the homomorphic image and pre image of anti Q-fuzzy subgroup of G with respect to t-conorm C is again an anti Q-fuzzy subgroup of G with respect to t-conorm C.

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