Fuzzy Normal Subgroups Research Articles

Subgroups and Homomorphism Structures of Complex Pythagorean Fuzzy Sets

This research introduces the notion of complex Pythagorean fuzzy subgroup (CPFSG). Both complex fuzzy subgroup (CFSG) and complex intuitionistic fuzzy subgroup (CIFSG) have significance in assigning membership grades in the unit disk in the complex plane. CFSG has a limitation solved by CIFSG, while CIFSG deals with a limited range of values. The important novelty of the CPFSG lies in its ability to solve the above limitations simultaneously and gets a wider range of values to be engaged in CPFSG. This work has introduced and investigated CPFSG as a new algebraic structure via the conditions that the sum of the square membership and non-membership lies on the unit interval for both the amplitude term and phase term. The result as any CIFSG is CPFSG but the convers is not true has been proved. Complex Pythagorean fuzzy coset has been defined and complex Pythagorean fuzzy normal subgroup (CPFNSG) and their algebraic characteristic has been demonstrated. Homomorphism on the CPFSG is shown. Some results as the inverse image of CPFSG and CPFNSG under isomorphism function are also a CPFSG and CPFNSG, respectively.

Roughness in Fuzzy Cayley Graphs

Rough set theory is a worth noticing approach for inexact and uncertain system modelling. When rough set theory accompanies with fuzzy set theory, which both are a complementary generalization of set theory, they will be attended by potency in theoretical discussions. In this paper a definition for fuzzy Cayley subsets is put forward as well as fuzzy Cayley graphs of fuzzy subsets on groups inspired from the definition of Cayley graphs. We introduce rough approximation of a Cayley graph with respect to a fuzzy normal subgroup. We introduce the approximation rough fuzzy Cayley graphs and fuzzy rough fuzzy Cayley graphs. The last approximation is the mixture of the other approximations. Some theorems and properties are investigated and proved.

Picture Fuzzy Subgroup

Picture fuzzy subgroup of a crisp group is established here and some properties connected to it are investigated. Also, normalized restricted picture fuzzy set, conjugate picture fuzzy subgroup, picture fuzzy coset, picture fuzzy normal subgroup and the order of picture fuzzy subgroup are defined. The order of picture fuzzy subgroup is defined using the cardinality of a special type of crisp subgroup. Some corresponding properties are established in this regard. Significant Statement. Subgroup is an important algebraic structure in the field of Pure Mathematics. Study of different properties of subgroup in fuzzy sense is an interesting fact to the readers because fuzzy sense is the extension of classical sense. Readers can easily observe how the properties of subgroup hold in fuzzy sense like classical sense. Picture fuzzy sense is the generalization of fuzzy sense. In other words, picture fuzzy sense can be treated as advanced fuzzy sense. Readers will be interested to study how the properties of subgroup hold when the number of components increases in fuzzy environment. Our study is actually the study of an important type of advanced fuzzy algebraic structure.

HOMOMORPHISM OF IMAF NORMAL SUBGROUP

ABSTRACT In this article an attempt has been made to study some new algebraic structures of intuitionistic multi-anti fuzzy normal subgroups under homomorphism are discussed. Keywords Intuitionistic multi-fuzzy set (IMFS), Intuitionistic multi-anti fuzzy subgroup (IMAFSG), Intuitionistic multi-anti fuzzy normal subgroup (IMAFNSG). Mathematics Subject Classification 20N25, 03E72, 08A72, 03F55, 06F35, 03G25, 08A05

ON THE NUMBER OF FUZZY SUBGROUPS AND FUZZY NORMAL SUBGROUPS OF S2, S3 AND A4

Counting fuzzy subgroups of a finite group is a fundamental problem of fuzzy group theory. Many researchers have made significant contributions to the rapid growth of this topic in recent years. The number of fuzzy subgroups of any group is infinite without the aid of equivalence relation. Some authors have used the equivalence relation of fuzzy sets to study the equivalence of fuzzy subgroups ([5], [6], [16]). The problem of counting the number of distinct fuzzy subgroups of a finite group is relative to the choice of the equivalence relation. The number of fuzzy subgroups of a particular group varies from one equivalence relation to the other. The equivalence relation applied in our computation can be seen in the existing literature. Sulaiman and Abd Ghafur [10] define an equivalence relation for counting fuzzy subgroups of group G. We have used this relation to find fuzzy subgroups and fuzzy normal subgroups of S2, S3 and A4. Lattice subgroup diagrams were used in our computation.

Characterization of Nested Lattice Codes Through Quotient Groups Relative to Normal Fuzzy Subgroups for Channel Quantization

Interference is in general, an undesirable obstacle to communication in wireless communication networks. Therefore, in order to realize interference alignment, this work intends to characterize classical codes through the framework of fuzzy set theory for quantizing real-valued channel coefficients. Thenceforward, from the classical framework nested real lattice codes are constructed by using construction A of nested real ideal lattices and, therefore, through normal fuzzy subgroups this work shows that the constructed quotient groups of the nested real ideal lattices relative to normal fuzzy subgroups of these nested real ideal lattices are nested vector spaces and, consequently, such nested quotient groups have the structure of a linear code, are isomorphic to the nested real lattice codes and defined by linear fuzzy codes. Through such an isomorphism and the construction of the nested quotient groups relative to normal fuzzy subgroups, it is possible to conclude that the codewords of these nested linear fuzzy codes are in a one-to-one correspondence to membership vectors. Through the constructed nested quotient groups, that is, nested linear fuzzy codes, this work supplies how to quantize real-valued channel coefficients onto these nested quotient groups in order to realize interference alignment and, consequently, it is possible to encode an arbitrary real-valued channel coefficient by a codeword from a linear fuzzy code.

(α1, 2, β1, 2)-complex intuitionistic fuzzy subgroups and its algebraic structure

<abstract> <p>A complex intuitionistic fuzzy set is a generalization framework to characterize several applications in decision making, pattern recognition, engineering, and other fields. This set is considered more fitting and coverable to Intuitionistic Fuzzy Sets (IDS) and complex fuzzy sets. In this paper, the abstraction of (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$) complex intuitionistic fuzzy sets and (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroups were introduced regarding to the concept of complex intuitionistic fuzzy sets. Besides, we show that (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. Also, each of (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy normal subgroups and cosets are defined and studied their relationship in the sense of the commutator of groups and the conjugate classes of group, respectively. Furthermore, some theorems connected the (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroup of the classical quotient group and the set of all (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy cosets were studied and proved. Additionally, we expand the index and Lagrange's theorem to be suitable under (${{\alpha _{1, 2}}, {\beta _{1, 2}}}$)-complex intuitionistic fuzzy subgroups.</p> </abstract>

A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms

The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean fuzzy subgroup is performed. We define the notions of Pythagorean fuzzy homomorphism and isomorphism and generalize the notion of factor group of a classical group W relative to its normal subgroup S by defining a PFSG of WS. At the end, the Pythagorean fuzzy version of fundamental theorems of isomorphisms is proved.

Algebraic Properties of ω , θ -Complex Fuzzy Subgroups

This paper defined the notion of ω , θ -complex fuzzy sets, ω , θ -complex fuzzy subgroupoid, and ω , θ -complex fuzzy subgroups and described important examples under ω , θ -complex fuzzy sets. Additionally, we discussed the conjugacy class of group with respect to ω , θ -complex fuzzy normal subgroups. We define ω , θ -complex fuzzy cosets and elaborate the certain operation of this analog to group theoretic operation. We prove that factor group with regard to ω , θ -complex fuzzy normal subgroup forms a group and establishes an ordinary homomorphism from group to its factor group with regard to ω , θ -complex fuzzy normal subgroup. Moreover, we create the ω , θ -complex fuzzy subgroup of factor group.

Some Characterizations of Certain Complex Fuzzy Subgroups

The complex fuzzy environment is an innovative tool to handle ambiguous situations in different mathematical problems. In this article, we commence the abstraction of (ρ,η)-complex fuzzy sets, (ρ,η)-complex fuzzy subgroupoid, (ρ,η)-complex fuzzy subgroups and describe important examples of the symmetric group under (ρ,η)-complex fuzzy sets. Additionally, we discuss the conjugacy class of the group with respect to (ρ,η)-complex fuzzy normal subgroups. We define (ρ,η)-complex fuzzy cosets and elaborate upon the certain operation of this analog to group theoretic operation. We prove that factors regarding the (ρ,η)-complex fuzzy normal subgroup form a group and establish an ordinary homomorphism. Moreover, we create the (ρ,η)-complex fuzzy subgroup of the factor group.

On Cubic Fuzzy Groups and Cubic Fuzzy Normal Subgroups

In this paper, the notions of cubic fuzzy groups and cubic fuzzy normal subgroups are introduced. The internal, external of cubic sets, (P-,R-) order, (P-,R-) intersection and (P-,R-) union of cubic fuzzy groups are investigated and some related properties were obtained. It is proved that a cubic fuzzy group which is both (internal, external) cubic set. Also we provide condition on cubic fuzzy group to be an internal cubic set. We show that (P-,R-) intersection and (P-,R-) union of cubic fuzzy groups are also cubic fuzzy groups. Also the (P-,R-) intersection, (P-,R-) union of cubic fuzzy normal subgroups are proved to be cubic fuzzy normal subgroup.

On q -Rung Orthopair Fuzzy Subgroups

The q -rung orthopair fuzzy environment is an innovative tool to handle uncertain situations in various decision-making problems. In this work, we characterize the idea of a q -rung orthopair fuzzy subgroup and examine various algebraic attributes of this newly defined notion. We also present q -rung orthopair fuzzy coset and q -rung orthopair fuzzy normal subgroup along with relevant fundamental theorems. Moreover, we introduce the concept of q -rung orthopair fuzzy level subgroup and proved related results. At the end, we explore the consequence of group homomorphism on the q -rung orthopair fuzzy subgroup.

On Fundamental Algebraic Characterizations ofμ-Fuzzy Normal Subgroups

In this article, we present the study ofμ-fuzzy subgroups and prove numerous fundamental algebraic attributes of this newly defined notion. We also define the concept ofμ-fuzzy normal subgroup and investigate many vital algebraic characteristics of these phenomena. In addition, we characterize the quotient group induced by this particular fuzzy normal subgroup and establish a group isomorphism between the quotient groupsG/κμandG/κ∗μ. Furthermore, we initiate the study of level subgroup, open level subgroup, and tangible subgroup of aμ-fuzzy subgroup and emphasize the significance ofμ-fuzzy normal subgroups by establishing a relationship between these newly defined notions andμ-fuzzy normal subgroup.

Homomorphism of Characteristic Fuzzy Subgroup and Abelian Fuzzy Subgroup

Abstract: In this paper, we have established some independent proof of homomorphism on algebra of abelian and characteristic fuzzy subgroup. The characteristic of fuzzy subgroup [13] was first introduced by P. Bhattacharya and N. P. Mukharjee in 1986. Keywords: Fuzzy subgroup, characteristic fuzzy subgroup, abelian fuzzy subgroup and normal fuzzy subgroup.

Certain Structure of Lagrange’s Theorem with the Application of Interval-Valued Intuitionistic Fuzzy Subgroups

This paper presents the concept of an interval-valued intuitionistic fuzzy subgroup defined on interval-valued intuitionistic fuzzy sets. We study some of the fundamental algebraic properties of interval-valued intuitionistic fuzzy cosets and interval-valued intuitionistic fuzzy normal subgroup of a given group. This idea is used to describe the interval-valued intuitionistic fuzzy order and index of interval-valued intuitionistic fuzzy subgroup. We have created numerous algebraic properties of interval-valued intuitionistic fuzzy order of an element. We also prove the interval-valued intuitionistic fuzzification of Lagrange’s theorem.

The Characterization of Substructures of γ‐Anti Fuzzy Subgroups with Application in Genetics

Fuzzy and anti fuzzy normal subgroups are the current instrument for dealing with ambiguity in various decision‐making challenges. This article discusses γ‐anti fuzzy normal subgroups and γ‐fuzzy normal subgroups. Set‐theoretic properties of union and intersection are examined and it is observed that union and intersection of γ‐anti fuzzy normal subgroups are γ‐anti fuzzy normal subgroups. Employee selection impacts the input quality of employees and hence plays an important part in human resource management. The cost of a group is established in proportion to the fuzzy multisets of a fuzzy multigroup. It was a good idea to introduce anti‐intuitionistic fuzzy sets and anti‐intuitionistic fuzzy subgroups, as well as to demonstrate some of their algebraic features. Product of γ‐anti fuzzy normal subgroups and γ‐fuzzy normal subgroups is defined, the product’s algebraic nature is analyzed, and the findings are supported by presenting γ‐anti typical sections with blurring and γ‐ordinary parts with the weirdness of well‐defined and well‐established groups of genetic codes.

\(S\)-norms on anti \(Q\)-fuzzy subgroups

In this paper, by using \(S\)-norms, we defined anti fuzzy subgroups and anti fuzzy normal subgroups which are new notions and considered their fundamental properties and also made an attempt to study the characterizations of them. Next we investigated image and pre image of them under group homomorphisms. Finally, we introduced the direct sum of them and proved that direct sum of any family of them is also anti fuzzy subgroups and anti fuzzy normal subgroups under \(S\)-norms, respectively.

Complex Fuzzy Groups Based on Rosenfeld’s Approach

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

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

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 -complex fuzzy sets and then define -complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an -complex fuzzy subgroup and define -complex fuzzy normal subgroups of given group. We extend this ideology to define -complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that -complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of -complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the -complex fuzzy subgroup of the classical quotient group and show that the set of all -complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of -complex fuzzy subgroups and investigate the -complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

The Number of Fuzzy Subgroups and Fuzzy Normal Subgroups of Dicyclic Group Q12

The Number of Fuzzy Subgroups and Fuzzy Normal Subgroups of Dicyclic Group Q12