scholarly journals Study on the Algebraic Structure of Refined Neutrosophic Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 954
Author(s):  
Qiaoyan Li ◽  
Yingcang Ma ◽  
Xiaohong Zhang ◽  
Juanjuan Zhang
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Multiplication Operator ◽  
Neutral Element ◽  
Algebra Structure ◽  
Multiplication Operation ◽  
General Field ◽  
Neutrosophic Number

This paper aims to explore the algebra structure of refined neutrosophic numbers. Firstly, the algebra structure of neutrosophic quadruple numbers on a general field is studied. Secondly, The addition operator ⊕ and multiplication operator ⊗ on refined neutrosophic numbers are proposed and the algebra structure is discussed. We reveal that the set of neutrosophic refined numbers with an additive operation is an abelian group and the set of neutrosophic refined numbers with a multiplication operation is a neutrosophic extended triplet group. Moreover, algorithms for solving the neutral element and opposite elements of each refined neutrosophic number are given.

scholarly journals Neutrosophic Extended Triplet Group Based on Neutrosophic Quadruple Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 696 ◽  
Author(s):  
Qiaoyan Li ◽  
Yingcang Ma ◽  
Xiaohong Zhang ◽  
Juanjuan Zhang
Keyword(s):  
Dimensional Space ◽  
Multiplication Operator ◽  
Neutral Element ◽  
Algebra Structure ◽  
Multiplication Operation ◽  
Group 2

In this paper, we explore the algebra structure based on neutrosophic quadruple numbers. Moreover, two kinds of degradation algebra systems of neutrosophic quadruple numbers are introduced. In particular, the following results are strictly proved: (1) the set of neutrosophic quadruple numbers with a multiplication operation is a neutrosophic extended triplet group; (2) the neutral element of each neutrosophic quadruple number is unique and there are only sixteen different neutral elements in all of neutrosophic quadruple numbers; (3) the set which has same neutral element is closed with respect to the multiplication operator; (4) the union of the set which has same neutral element is a partition of four-dimensional space.

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

Hopf Algebras from Branching Rules

1983 ◽  
Vol 35 (1) ◽  
pp. 177-192 ◽  
Author(s):  
P. Hoffman
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Homogeneous Element ◽  
Algebra Structure ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Dimension Zero ◽  
Branching Rules

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.

scholarly journals A Study on Finite Abelian Groups: Sylow’s Theorems Based

2017 ◽  
Vol 1 (3) ◽  
Author(s):  
Amaira Moaitiq Mohammed Al-Johani
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Abstract Algebra ◽  
Vector Spaces ◽  
Finite Abelian Groups ◽  
Study Objective ◽  
Group Operation ◽  
Basic Concepts

In abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms. Algebraic structures include groups, rings, fields, and lattices, etc. A group is an algebraic structure (????, ∗), which satisfies associative, identity and inverse laws. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutatively. The concept of an Abelian group is one of the first concepts encountered in abstract algebra, from which many other basic concepts, such as rings, commutative rings, modules and vector spaces are developed. This study sheds the light on the structure of the finite abelian groups, basis theorem, Sylow’s theorem and factoring finite abelian groups. In addition, it discusses some properties related to these groups. The researcher followed the exploratory and comparative approaches to achieve the study objective. The study has shown that the theory of Abelian groups is generally simpler than that of their non-abelian counter parts, and finite Abelian groups are very well understood.  

scholarly journals Electronic Module Development on Algebra Structure Courses

2020 ◽  
Vol 3 (3) ◽  
pp. 316-328
Author(s):  
Olympia Agustina ◽  
Farida Farida ◽  
Fredi Ganda Putra
Keyword(s):  
Mathematics Education ◽  
Algebraic Structure ◽  
Quantitative Data ◽  
Qualitative Data ◽  
Development Model ◽  
Algebra Structure ◽  
Due Diligence ◽  
Education Students ◽  
Student Responses ◽  
Module Development

This study aims to describe the process of developing an electronic module in the algebraic structure course, and knowing the effectiveness of the electronic module after being tested on mathematics education students. This research was conducted in class C, 5th semester, majoring in mathematics education at UIN Raden Intan Lampung. The development model used in this research is the Borg and Gall development model. The development of the electronic module was carried out based on the Articulate Studio'13 application. The types of data taken in this study were qualitative data and quantitative data. From the results of due diligence from experts and student responses, the results of the development of an electronic module are very feasible and very interesting. Then, based on the results of the final assessment of student’s effectiveness of the product showed a presentation of more than 75%. Based on the research results, it can be concluded that the electronic module in the algebraic structure course is effectively used by mathematics education students.

scholarly journals A note on almost prime submodule of CSM module over principal ideal domain

2021 ◽  
Vol 2106 (1) ◽  
pp. 012011
Author(s):  
I G A W Wardhana ◽  
N D H Nghiem ◽  
N W Switrayni ◽  
Q Aini
Keyword(s):  
Prime Ideal ◽  
Algebraic Structure ◽  
Principal Ideal ◽  
Ring Theory ◽  
Principal Ideal Domain ◽  
Module Theory ◽  
Ideal Domain ◽  
Multiplication Operation ◽  
Prime Submodule ◽  
Almost Prime

Abstract An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

scholarly journals Galkin Quandles, Pointed Abelian Groups, and Sequence A000712

10.37236/2676 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
William E. Clark ◽  
Xiang-dong Hou
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Knot Invariants ◽  
Group A

For each pointed abelian group $(A,c)$, there is an associated Galkin quandle $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.

scholarly journals PEMAHAMAN KONSEP MAHASISWA TENTANG RING PADA MATA KULIAH STRUKTUR ALJABAR 2 DITINJAU DARI PEMIKIRAN KREATIF PADA SISWA KELOMPOK ATAS

2017 ◽  
Vol 2 (2) ◽  
pp. 69
Author(s):  
Maya Rini Rubowo ◽  
FX. Didik Purwosetiyono ◽  
Dewi Wulandari
Keyword(s):  
Qualitative Research ◽  
Algebraic Structure ◽  
Creative Thinking ◽  
Identity Element ◽  
Ring Theory ◽  
Algebra Structure ◽  
Math Skills ◽  
Recording Equipment ◽  
Mathematics Student ◽  
The Subject

This research is intended to describe the students concept of the ring in the course of algebra structure 2 in terms of creative thinking in upper students. This research is included in this type of qualitative research. The subject of the study is a mathematics student who has taken the course of Structure Algebra 2 (Ring Theory). The subjects in this study are three students who have high math skills. Instrument  used in research 1) Test of Concept of Ring Concept (TPKR), 2) Interview Guidelines, 3) Creative Thinking Rubric, 4) Recording Equipment, 5) Documentation. Based on the results of the study and discussion, the findings involving the students' concept of the ring in the course of algebraic structure 2 in the upper sstudent can solve the problem in terms of explaining the associative nature, explaining the nature of the identity element, explaining the nature of each element having a good inverse. However, I can not only use one solution only, but it is less diverging and able to find other solutions. At the student level still not able to meet the original indicators in creative thinking

Scalar Actions

1983 ◽  
Vol 35 (4) ◽  
pp. 750-768
Author(s):  
A. Lebow ◽  
M. Schreiber
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Ring Homomorphism ◽  
Space Structure ◽  
Point Of View ◽  
Identity Operator ◽  
Representation Space ◽  
New Space ◽  
The Subject ◽  
Scalar Action

The subject of this paper arises from the familiar process whereby an automorphism of a field generates new representations from old. One may think of that process spatially, as a change of vector space structure in the representation space by means of the automorphism. The operators of the representation acting in the “new“ space then constitute the new representation. This point of view makes visible an algebraic structure we call a scalar action. A scalar action f of a ring R (with unity) in an abelian group Kis a ring homomorphism f:R → End(V) taking the unity element of R to the identity operator in End(V). If f is a scalar action of a field F and ϕ is an automorphism of F then f ∘ ϕ is another scalar action of F, and it is this construction which is used to define the “new” representation space mentioned above. But the variety of scalar actions goes rather beyond that construction.

scholarly journals Retracts of trees and free left adequate semigroups

2011 ◽  
Vol 54 (3) ◽  
pp. 731-747 ◽  
Author(s):  
Mark Kambites
Keyword(s):  
Algebraic Structure ◽  
Free Object ◽  
Multiplication Operation ◽  
Algebraic Interpretation ◽  
Free Left

AbstractRecent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups.

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