Fuzzy Group Theory Research Articles

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.

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.

Intuitionistic Level Subgroups in the Dihedral Group D3

A well known result in fuzzy group theory states that “level subgroups of any fuzzy subgroup of a finite group form a chain”. We check the validity of this statement in the intuitionistic fuzzy perspective. We do this using Dihedral Group $D_3$, which is a non-cyclic group. We prove that $D_3$ has 100 distinct intuitionistic fuzzy subgroups (IFSGs) upto isomorphism. The intuitionistic level subgroups (ILSGs) of exactly 76 among them make chains, and hence it can be concluded that the result is not true in the intuitionistic fuzzy perspective. We also enlist all the 100 distinct intuitionistic fuzzy subgroups of $D_3$ upto isomorphism.

On the number of fuzzy subgroups of dicyclic groups

One of the significant aspects of fuzzy group theory is classification of the fuzzy subgroups of finite groups under a suitable equivalence relation. In this paper, we determine the number of distinct fuzzy subgroups of dicyclic groups in some particular cases by the new equivalence relation introduced by Tǎrnǎuceanu. In this case, the corresponding equivalence classes of fuzzy subgroups of a group G are closely connected to the automorphism group and the chains of subgroups of G. In fact, this new equivalence relation generalizes the natural equivalence relation defined on the lattice of fuzzy subgroups.

Counting Distinct Fuzzy Subgroups of Symmetric Group S5 by a New Equivalence Relation

One of the significant aspects of fuzzy group theory is a classification of the fuzzy subgroups of finite groups under a suitable equivalence relation. In this paper, we determine the number of distinct fuzzy subgroups of finite symmetric group S5 by the new equivalence relation introduced by Tărnăuceanu. In this case, the corresponding equivalence classes of fuzzy subgroups of a group G are closely connected to the automorphism group and the chains of subgroups of G.

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

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.

Frattini fuzzy subgroups of fuzzy groups

This paper continues the study of fuzzy group theory which has been explored over times. We propose maximal fuzzy subgroups and Frattini fuzzy subgroups of fuzzy groups as extensions of maximal subgroups and Frattini subgroups of classical groups. It is shown that every Frattini fuzzy subgroup is both characteristic and normal,respectively. Finally, some results are established in connection to level subgroups and alpha cuts of fuzzy groups.

The Algebraic Structures of Complex Intuitionistic Fuzzy Soft Sets Associated with Groups and Subgroups

In recent years, the theory of complex fuzzy sets has captured the attention of many researchers, and research in this area has intensified in the past five years. However, almost all of the researchers in this area have focused on the development of various complex fuzzy based models as well as constructing decision making processes using current decision making approaches and tools. In this spirit, this paper focuses on developing the algebraic structures pertaining to groups and subgroups for the complex intuitionistic fuzzy soft set model. This paper was constructed based on the complex intuitionistic fuzzy soft set model which is characterized by a membership and a non-membership structure for both the amplitude and phase terms of the elements. This model was chosen due to its dual-membership structure that is better able to handle the uncertainties and partial ignorance that exists in most complex data, whilst retaining all the characteristics and advantages of complex fuzzy sets. Besides examining the properties and structural characteristics of the algebraic structures, the relationship between the algebraic structures introduced here and the corresponding algebraic structures in fuzzy group theory and classical group theory were also discussed and verified.

On the converse of Fuzzy Lagrange's Theorem

In fuzzy group theory many versions of the well-known Lagrange’s theorem have been studied. The aim of this article is to investigate the converse of one of those results. This leads to an interesting characterization of finite cyclic groups.

Structures on Fuzzy Groups and L- Fuzzy Number

The theory of fuzzy sets has developed in many directions and is finding applications in a wide variety of fields. Resentful in 1971 used this concept to develop the theory of fuzzy groups in this paper we have given independent proof of several theorems on fuzzy groups. We discuss about Fuzzy groups and investigate some of their structures on the concept of Fuzzy Group family and L- fuzzy number.

Fuzzy group theory: A comparison of different notions of product of fuzzy sets

The notion of penultimate subgroup of a fuzzy group is introduced. This leads to the concept of penultimate product of fuzzy subgroups. A detailed study of different notions of product of fuzzy sets in a groupoid is carried out. The ultimate superiority of Liu's set product is established. Level subsets and strong level subsets of these products are also analysed. Consequently, simplified proofs of some of the results on free product are obtained.

Fuzzy orders relative to fuzzy subgroups

In this paper, we introduce the notion of fuzzy orders of the elements of a group and show that most of the basic properties of orders of elements in the group theory are valid in the theory of fuzzy groups where ordinary orders of elements are replaced by fuzzy orders of elements. Reports of further development of the subject will follow in subsequent essays.

On fuzzy spaces and fuzzy group theory

The concept of fuzzy space is introduced. This concept corresponds to the concept of the universal set in the ordinary case. The algebra of fuzzy spaces and fuzzy subspaces is studied. Using the concept of fuzzy space and fuzzy binary operation, a new approach to study fuzzy group theory is given. This approach can be considered as a generalization and reformulation of the Rosenfeld theory of fuzzy subgroups. Therefore, it is an active tool to develop the theory of fuzzy groups.

Fuzzy groups with operators

Abstract In this paper, we further develop the fuzzy group theory based on Rosenfeld, 1971; Wu, 1981 and introduce some concepts such as fuzzy group with operators, fuzzy normal subgroup with operators, fuzzy factor group, etc. We also study their fundamental qualities and obtain some results.

Fuzzy subgroups: Some characterizations

The notion of a fuzzy set was first introduced in 1965 by Zadeh [12]. Since then the theory of fuzzy sets has gone through remarkably rapid strides with well over 4000 papers by now (see, for example, the bibliography [3]), several textbooks (for example, [2 and 6]), and an international journal solely devoted to it [S]. The theory has found wideranging applications in such diverse fields as automata, control theory, decision theory, and social behavior pattern studies, to name just a few. For an interesting, expository account see Rosenfeld [lo], and also the book review [7]. In [9], Rosenfeld introduced the notion of a fuzzy group and showed that many concepts of group theory can be extended in an elementary manner to develop the theory of fuzzy groups. In particular, he characterized all the fuzzy subgroups of cyclic groups of prime order [9, Proposition 5.101. Fuzzy groups were further investigated by Das in [1] where he characterized all the fuzzy subgroups of finite cyclic groups [l, Theorem 4.21. In this connection, he introduced the notion of “level subgroups” of a fuzzy subgroup which is based on the notion of a “level subset” introduced earlier by Zadeh. For a finite group Das showed that these level subgroups of a fuzzy subgroup form a chain [ 1, Corollary 3.11 and also constructed a fuzzy subgroup whose chain of level subgroups is maximal in the following special cases when the group is: (i) a direct product of cyclic groups of prime orders, (ii) abelian [l, Theorem 3.41. We analyze the level subgroups of a fuzzy subgroup in more detail and investigate whether the family of level subgroups of a fuzzy subgroup deter-