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

2021 ◽  
Vol 4 (3) ◽  
pp. 1-9
Author(s):  
Rasul Rasuli ◽  
Keyword(s):  
Normal Subgroups ◽  
Fundamental Properties ◽  
Direct Sum ◽  
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.

scholarly journals Equivalent Expressions of Direct Sum Decomposition of Groups

2015 ◽  
Vol 23 (1) ◽  
pp. 67-73
Author(s):  
Kazuhisa Nakasho ◽  
Hiroyuki Okazaki ◽  
Hiroshi Yamazaki ◽  
Yasunari Shidama
Keyword(s):  
Equivalent Form ◽  
Normal Subgroups ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Equivalent Expressions

Summary In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the formalization.

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.

The lattices of fuzzy subgroups and fuzzy normal subgroups

1994 ◽  
Vol 76 (1-2) ◽  
pp. 1-11 ◽  
Author(s):  
Naseem Ajmal ◽  
K.V. Thomas
Keyword(s):  
Normal Subgroups ◽  
Fuzzy Subgroups

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.

Representation and central extension of hom-Lie algebroids

2018 ◽  
Vol 17 (11) ◽  
pp. 1850219 ◽  
Author(s):  
S. Merati ◽  
M. R. Farhangdoost
Keyword(s):  
Central Extension ◽  
Lie Algebroids ◽  
Central Extensions ◽  
Fundamental Properties ◽  
Direct Sum ◽  
One To One

The aim of this paper is to develop the theory of representation of hom-Lie algebroids. After introducing some key constructions and examples of hom-Lie algebroids involving sub-Lie hom-algebroids and direct sum hom-Lie algebroids, we describe the notion and some properties of infinitesimal action of hom-Lie algebroids. We introduce concept of representation of hom-Lie algebroids and prove some fundamental properties to show a one to one correspondence between representations and exterior differentials. Finally, we review trivial representations and its associated cohomology to introduce the central extensions. Also, we show that the central extensions induced by two trivial representations are isomorphic if their [Formula: see text]-forms are cohomologous.

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.

scholarly journals A Study of(λ,μ)-Fuzzy Subgroups

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yuying Li ◽  
Xuzhu Wang ◽  
Liqiong Yang
Keyword(s):  
Normal Subgroups ◽  
Basic Properties ◽  
Fuzzy Subgroups

We deal with topics regarding(λ,μ)-fuzzy subgroups, mainly(λ,μ)-fuzzy cosets and(λ,μ)-fuzzy normal subgroups. We give basic properties of(λ,μ)-fuzzy subgroups and present some results related to(λ,μ)-fuzzy cosets and(λ,μ)-fuzzy normal subgroups.

scholarly journals Fuzzy Equivalence Relation, Fuzzy Congrunce Relation and Fuzzy Normal Subgroups on Group G Over T-Norms

2019 ◽  
Vol 7 (2) ◽  
Author(s):  
Rasul Rasuli
Keyword(s):  
Direct Product ◽  
Equivalence Relation ◽  
Fuzzy Relation ◽  
Normal Subgroups ◽  
Fuzzy Subgroups

In this study, by using t-norms, fuzzy equivalence relation, fuzzy congrunce relation on group G, fuzzy relation of subgroup H of group G, fuzzy normal subgroups of fuzzy subgroups, direct product of fuzzy subgroups(normal fuzzy subgroups) are introduced and some the their properties will be discussed. Next by using group homomorphisms, the image and pree image of them will be investigated.

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