Intuitionistic Fuzzy Subgroup Research Articles

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

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

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

(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>

On Intuitionistic Fuzzy Subgroups of (M-N) Type and Their Algebraic Properties

The objective of this paper is to define the intuitionistic fuzzy subgroup of type (M-N) and to study some of its elementary properties and substructures such as normality, direct and inverse images.Also, many related theorems and examples will be presented and illustrated.

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.

Homomorphisms of Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3.

The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the L-fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L, and lattice homomorphism from L to the interval [0,1]. In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.

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.

Algebraic properties of implication-based intuitionistic fuzzy finite state machine over a finite group

An extensive study is performed on the already constituted algebraic structure called the implication-based intuitionistic fuzzy semiautomaton (IB-IFSA) over a finite group that compromises of implication-based intuitionistic fuzzy kernel and implication-based intuitionistic fuzzy subsemiautomaton. In this paper, the necessary and sufficient condition for an implication-based intuitionistic fuzzy normal subgroup to be an implication-based intuitionistic fuzzy kernel of an implication-based intuitionistic fuzzy semiautomaton is developed. In addition, the fundamental and adequate condition for an implication-based intuitionistic fuzzy subgroup to be an implication-based intuitionistic fuzzy subsemiautomaton of an IB-IFSA is formulated. Moreover, the condition for an implication-based intuitionistic fuzzy kernel of the an IB-IFSA to be implication-based intuitionistic fuzzy subsemiautomaton is also established.

On the characterization of Pythagorean fuzzy subgroups

Pythagorean fuzzy environment is the modern tool for handling uncertainty in many decisions making problems. In this paper, we represent the notion of Pythagorean fuzzy subgroup (PFSG) as a generalization of intuitionistic fuzzy subgroup. We investigate various properties of our proposed fuzzy subgroup. Also, we introduce Pythagorean fuzzy coset and Pythagorean fuzzy normal subgroup (PFNSG) with their properties. Further, we define the notion of Pythagorean fuzzy level subgroup and establish related properties of it. Finally, we discuss the effect of group homomorphism on Pythagorean fuzzy subgroup.

On some algebraic aspects of η-intuitionistic fuzzy subgroups

In this study, we presents the idea of -intuitionistic fuzzy subgroup (IFSG) defined on -intuitionistic fuzzy set (IFS). Furthermore, we prove that every IFSG is an -IFSG. Also, we extend the study of this notion to define -intuitionistic fuzzy cosets and -intuitionistic fuzzy normal subgroups of a given group and investigate some of their fundamental algebraic features. Besides, we define the -intuitionistic fuzzy homomorphism between two -IFSG’s and show that an -intuitionistic fuzzy homomorphic image (inverse image) of the -IFSG is an -IFSG.

Algebraic Characteristics of Anti-Intuitionistic Fuzzy Subgroups Over a Certain Averaging Operator

In this paper, we propose the concept of ρ anti-intuitionistic fuzzy sets, ρ anti - intuitionistic fuzzy subgroups and prove some of their algebraic properties. We investigate a necessary and sufficient condition for a ρ-anti intuitionistic fuzzy set to be a ρ-anti intuitionistic fuzzy subgroup. We extend this ideology by defining the notions of ρ anti-intuitionistic fuzzy coset, ρ anti-intuitionistic fuzzy normal subgroup and derive some of their key algebraic characteristics. In addition, we study the quotient group of a group induced by ρ-anti intuitionistic fuzzy normal subgroup and establish a group isomorphism between this newly defined quotient group and the quotient group of group G relative to its particular normal subgroup G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sρ</sub> .

A Certain Class of t-Intuitionistic Fuzzy Subgroups

In this study, the t-intuitionistic fuzzy normalizer and centralizer of t intuitionistic fuzzy subgroup are proposed. The t-intuitionistic fuzzy centralizer is normal subgroup of t-intuitionistic fuzzy normalizer and investigate various algebraic properties of this phenomena. We also introduce the concept of t-intuitionistic fuzzy Abelian and cyclic subgroups and prove that every t-intuitionistic fuzzy subgroup of Abelian (cyclic) group is t-intuitionistic fuzzy Abelian (cyclic) subgroup. We show that the image and pre-image of t-intuitionistic fuzzy Abelian (cyclic) subgroup are t-intuitionistic fuzzy Abelian (cyclic) subgroup under group homomorphism.

A Novel Applications of Complex Intuitionistic Fuzzy Sets in Group Theory

In this paper, we initiate the novel concept of complex intuitionistic fuzzy subgroups and prove that every complex intuitionistic fuzzy subgroup generates two intuitionistic fuzzy subgroups. We extend this ideology to define the concept of level subsets of complex intuitionistic fuzzy set and discuss its various fundamental algebraic characteristics. We also show that the level subset of the complex intuitionistic fuzzy subgroups is a subgroup. Furthermore, we investigate the homomorphic image and pre-image of complex intuitionistic fuzzy subgroup under group homomorphism. Moreover, we prove that the product of two complex intuitionistic fuzzy subgroups is also a complex intuitionistic fuzzy subgroup and develop some new results about direct product of complex intuitionistic fuzzy subgroups.

Product of Implication-Based Intuitionistic Fuzzy Semiautomatons over Finite Groups

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.

A Study on Intuitionistic Multi-Anti Fuzzy Subgroups

A Study on Intuitionistic Multi-Anti Fuzzy Subgroups

(λ, μ)-Intuitionistic Fuzzy Subgroups of Groups with Operators

(λ, μ)-Intuitionistic Fuzzy Subgroups of Groups with Operators

English

The notion of intuitionistic fuzzy abelian subgroup and cyclic intuitionistic fuzzy subgroups are proposed, and then their properties and relations are discussed.

The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets

In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of λ-cut sets of IFS and show that λ-cut sets of IFS have the same properties as the cut sets of Zadeh fuzzy set. Thirdly, by using the concepts of λ-cut set of IFS and INS, we build up the decomposition theorem, representation theorem and extension principle of IFS. Finally, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup.

Interval-valued intuitionistic fuzzy subgroups based on interval-valued double t-norm

In this paper, the interval-valued double t-norm \(\widetilde{T}\) induced by an interval t-norm \(\overline{T}\) and an interval t-conorm \(\overline{S}\) is defined. The interval-valued intuitionistic fuzzy subgroup and the interval-valued intuitionistic fuzzy normal subgroup based on the interval-valued double t-norm \(\widetilde{T}\) are introduced. In the meantime, some of its basic properties and structural characteristics are studied. Furthermore, the theorems of the homomorphic image and the inverse image are given.

On the definition of the intuitionistic fuzzy subgroups

In this paper, a new kind of intuitionistic fuzzy subgroup theory, which is different from that of Ma, Zhan and Davvaz (2008) [22], [23], is presented. First, based on the concept of cut sets on intuitionistic fuzzy sets, we establish the neighborhood relations between a fuzzy point xa and an intuitionistic fuzzy set A. Then we give the definitions of the grades of xa belonging to A, xa quasi-coincident with A, xa belonging to and quasi-coincident with A and xa belonging to or quasi-coincident with A, respectively. Second, by applying the 3-valued Lukasiewicz implication, we give the definition of (α,β)-intuitionistic fuzzy subgroups of a group G for α,β∈{∈,q,∈∧q,∈∨q}, and we show that, in 16 kinds of (α,β)-intuitionistic fuzzy subgroups, the significant ones are the (∈,∈)-intuitionistic fuzzy subgroup, the (∈,∈∨q)-intuitionistic fuzzy subgroup and the (∈∧q,∈)-intuitionistic fuzzy subgroup. We also show that A is a (∈,∈)-intuitionistic fuzzy subgroup of G if and only if, for any a∈(0,1], the cut set Aa of A is a 3-valued fuzzy subgroup of G, and A is a (∈,∈∨q)-intuitionistic fuzzy subgroup (or (∈,∈∨q)-intuitionistic fuzzy subgroup) of G if and only if, for any a∈(0,0.5](or for any a∈(0.5,1]), the cut set Aa of A is a 3-valued fuzzy subgroup of G. At last, we generalize the (∈,∈)-intuitionistic fuzzy subgroup, (∈,∈∨q)-intuitionistic fuzzy subgroup and (∈∧q,∈)-intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., (s,t]-intuitionistic fuzzy subgroups. We show that A is a (s,t]-intuitionistic fuzzy subgroup of G if and only if, for any a∈(s,t], the cut set Aa of A is a 3-valued fuzzy subgroup of G. We also characterize the (s,t]-intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point xa and an intuitionistic fuzzy set A.