scholarly journals Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 321 ◽  
Author(s):  
Mehmet Çelik ◽  
Moges Shalla ◽  
Necati Olgun
Keyword(s):  
Group Theory ◽  
Significant Role ◽  
Classical Group ◽  
Algebraic Structures ◽  
Algebraic Systems ◽  
Homomorphism Theorem ◽  
Special Cases ◽  
Isomorphism Theorems

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

scholarly journals Norton Algebras of the Hamming Graphs via Linear Characters

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Nonassociative Algebra ◽  
Bilinear Forms ◽  
Distance Regular Graph ◽  
Finite Abelian Groups ◽  
Hamming Graph ◽  
Large Subgroup ◽  
Special Cases ◽  
Hamming Graphs

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

scholarly journals Applications of Soft Union Sets in the Ring Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yongwei Yang ◽  
Xiaolong Xin ◽  
Pengfei He
Keyword(s):  
Ring Theory ◽  
Algebraic Structures ◽  
Soft Sets ◽  
Quotient Rings ◽  
Isomorphism Theorems ◽  
Algebraic Tool

The aim of the paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion ofλ,μ-soft union rings which is a generalization of that of soft union rings is proposed. By introducing the notion of soft cosets, soft quotient rings based onλ,μ-soft union ideals are established. Moreover, through discussing quotient soft subsets, an approach for constructing quotient soft union rings is made. Finally, isomorphism theorems ofλ,μ-soft union rings related to invariant soft sets are discussed.

A computer proof of relations in a certain class of groups

1991 ◽  
Vol 117 (1-2) ◽  
pp. 109-114
Author(s):  
Edmund F. Robertson ◽  
Kevin Rutherford
Keyword(s):  
Group Theory ◽  
Special Cases ◽  
Computer Proof

SynopsisA gp-toolkit consisting of computer implementations of various group theory methods, in particular a Tietze transformation program, was designed. Special cases of a conjecture were solved by the gp-toolkit. Examination of the method used by the gp-toolkit to deduce relations showed that a general approach had been employed. We present a proof verifying that the conjecture is true which is a straightforward generalisation of the method discovered by the gp-toolkit.

Some properties of repetends

2003 ◽  
Vol 87 (510) ◽  
pp. 437-443
Author(s):  
N. J. Armstrong ◽  
R. J. Armstrong
Keyword(s):  
Group Theory ◽  
General Condition ◽  
Number System ◽  
Special Cases ◽  
The Subject

We wish to discuss some aspects of repetends, the repeating sequence of digits in the expansion of a fraction (for illuminating introductions to the subject see [1, 2]). For the most part we restrict consideration here to fractions with a prime denominator. But we do consider the general condition for the length of repetends and examine some special cases when the base of the number system is varied. An illustration of the use of other bases than 10 is given. Then we consider the multiplication of repetends and show a connection with group theory, giving an old result by a new twist.

scholarly journals Some relations and subsets generated by principal consistent subset of semigroup with apartness

2002 ◽  
pp. 7-25 ◽  
Author(s):  
Abraham Romano
Keyword(s):  
Set Theory ◽  
Semigroup Theory ◽  
Left Zero ◽  
Constructive Mathematics ◽  
Algebraic Structures ◽  
Main Subject ◽  
Special Cases ◽  
Constructive Algebra ◽  
Consistent Subset ◽  
Semigroup With Apartness

The investigation is in the Constructive algebra in the sense of E. Bishop, F. Richman, W. Ruitenburg, D. van Dalen and A. S. Troelstra. Algebraic structures with apartness the first were defined and studied by A. Heyting. After that, some authors studied algebraic structures in constructive mathematics as for example: D. van Dalen, E. Bishop, P. T. Johnstone, A. Heyting, R. Mines, J. C. Mulvey, F. Richman, D. A. Romano, W. Ruitenburg and A. Troelstra. This paper is one of articles in their the author tries to investigate semugroups with apartnesses. Relation q on S is a coequality relation on S if it is consistent, symmetric and cotran-sitive; coequality relation is generalization of apatness. The main subject of this consideration are characterizations of some coequality relations on semigroup S with apartness by means od special ideals J(a) = {x E S : a# SxS}, principal consistent subsets C(a) = {x E S : x# SaS} (a E S) of S and by filled product of relations on S. Let S = (S, =, 1) be a semigroup with apartness. As preliminaries we will introduce some special notions, notations and results in set theory, commutative ring theory and semigroup theory in constructive mathematics and we will give proofs of several general theorems in semigroup theory. In the next section we will introduce relation s on S by (x, y) E s iff y E C(x) and we will describe internal filfulments c(s U s?1) and c(s ? s?1) and their classes A(a) = ?An(a) and K(a) = ?Kn(a) respectively. We will give the proof that the set K(a) is maximal strongly extensional consistent ideal of S for every a in S. Before that, we will analyze semigroup S with relation q = c(s U s?1 ) in two special cases: (i) the relation q is a band coequality relation on S : (ii) q is left zero band coequality relation on S. Beside that, we will introduce several compatible equality and coequality relations on S by sets A(a), An(a), K(a) and Kn(a).

scholarly journals The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 411
Author(s):  
Faisal Mehmood ◽  
Fu-Gui Shi ◽  
Khizar Hayat ◽  
Xiao-Peng Yang
Keyword(s):  
Group Theory ◽  
Continuous Functions ◽  
Vital Role ◽  
Ring Theory ◽  
Mathematical Tool ◽  
Algebraic Structures ◽  
Ring Homomorphisms ◽  
Extremum Problems ◽  
Convexity Theory ◽  
Fundamental Theorems

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

Maximal and Frattini L-subgroups of an L-group

2020 ◽  
Vol 39 (3) ◽  
pp. 3995-4007
Author(s):  
I. Jahan ◽  
A. Manas
Keyword(s):  
Group Theory ◽  
Classical Group

In this paper, the concept of a maximal L-subgroup of an L-group has been defined in the spirit of classical group theory. Then, a level subset characterization has been established for the same. Then, this notion of maximal L-subgroups has been used to define Frattini L-subgroup. Further, the concept of non-generators of an L-group has been developed and its relation with the Frattini L-subgroup of an L-group has been established like their classical counterparts. Moreover, several properties pertaining to the concepts of maximal L-subgroups and Frattini L-subgroup have also been investigated. These two notions have been illustrated through several examples.

Some Group Theoretic Notions in Fuzzy Multigroup Context

2020 ◽  
pp. 34-62 ◽  
Author(s):  
Paul Augustine Ejegwa
Keyword(s):  
Group Theory ◽  
Direct Product ◽  
Classical Group ◽  
The Relationship ◽  
Product Of Groups

The notion of fuzzy multigroups is an application of fuzzy multisets or fuzzy bags to classical group theory. This chapter explores the notions of fuzzy comultisets, characteristic fuzzy submultigroups, and direct product of fuzzy multigroups as extensions of cosets, characteristic subgroups, and direct product of groups. The relationship between fuzzy comultisets of a fuzzy multigroup and the cosets of a group is established. Some results on the concept of fuzzy comultisets are deduced. A number of properties of characteristic fuzzy submultigroups of fuzzy multigroups are outlined, and some related results are obtained. Also, the author presents some properties of direct product of fuzzy multigroups and establish some results.

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