scholarly journals Study of Two Kinds of Quasi AG-Neutrosophic Extended Triplet Loops

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaogang An ◽  
Mingming Chen
Keyword(s):  
Structural Characteristics ◽  
Algebraic Structures ◽  
Basic Properties ◽  
Generalized Symmetries

Abel-Grassmann’s groupoid and neutrosophic extended triplet loop are two important algebraic structures that describe two kinds of generalized symmetries. In this paper, we investigate quasi AG-neutrosophic extended triplet loop, which is a fusion structure of the two kinds of algebraic structures mentioned above. We propose new notions of AG-(l,r)-Loop and AG-(r,l)-Loop, deeply study their basic properties and structural characteristics, and prove strictly the following statements: (1) each strong AG-(l,r)-Loop can be represented as the union of its disjoint sub-AG-groups, (2) the concepts of strong AG-(l,r)-Loop, strong AG-(l,l)-Loop, and AG-(l,lr)-Loop are equivalent, and (3) the concepts of strong AG-(r,l)-Loop and strong AG-(r,r)-Loop are equivalent.

scholarly journals A Kind of Variation Symmetry: Tarski Associative Groupoids (TA-Groupoids) and Tarski Associative Neutrosophic Extended Triplet Groupoids (TA-NET-Groupoids)

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 714
Author(s):  
Xiaohong Zhang ◽  
Wangtao Yuan ◽  
Mingming Chen ◽  
Florentin Smarandache
Keyword(s):  
Associative Ring ◽  
Disjoint Union ◽  
Structural Characteristics ◽  
Basic Properties ◽  
Generalized Symmetries ◽  
Associative Law

The associative law reflects symmetry of operation, and other various variation associative laws reflect some generalized symmetries. In this paper, based on numerous literature and related topics such as function equation, non-associative groupoid and non-associative ring, we have introduced a new concept of Tarski associative groupoid (or transposition associative groupoid (TA-groupoid)), presented extensive examples, obtained basic properties and structural characteristics, and discussed the relationships among few non-associative groupoids. Moreover, we proposed a new concept of Tarski associative neutrosophic extended triplet groupoid (TA-NET-groupoid) and analyzed related properties. Finally, the following important result is proved: every TA-NET-groupoid is a disjoint union of some groups which are its subgroups.

scholarly journals A Novel Approach towards Bipolar Soft Sets and Their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Tahir Mahmood
Keyword(s):  
Decision Making ◽  
Algebraic Structures ◽  
Soft Sets ◽  
New Approach ◽  
Basic Properties ◽  
Novel Approach ◽  
Operational Laws

The notion of bipolar soft sets has already been defined, but in this article, the notion of bipolar soft sets has been redefined, called T-bipolar soft sets. It is shown that the new approach is more close to the concept of bipolarity as compared to the previous ones, and further it is discussed that so far in the study of soft sets and their generalizations, the concept introduced in this manuscript has never been discussed earlier. We have also discussed the operational laws of T-bipolar soft sets and their basic properties. In the end, we have deliberated the algebraic structures associated with T-bipolar soft sets and the applications of T-bipolar soft sets in decision-making problems.

Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ

2005 ◽  
Vol 42 (4) ◽  
pp. 371-386
Author(s):  
M. Aslam Malik ◽  
S. M. Husnine ◽  
Abdul Majeed
Keyword(s):  
Quadratic Field ◽  
Infinite Group ◽  
Algebraic Structures ◽  
Real Quadratic Field ◽  
Irrational Numbers ◽  
The Real ◽  
Basic Properties ◽  
Quadratic Irrational ◽  
Real Quadratic

Studying groups through their actions on different sets and algebraic structures has become a useful technique to know about the structure of the groups. The main object of this work is to examine the action of the infinite group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H = \langle x,y : x^{2} = y^{4} = 1\rangle$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $x (z) = \frac{-1}{2z}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $y (z) = \frac{-1}{2(z+1)}$ \end{document} on the real quadratic field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} and find invariant subsets of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H$ \end{document}. We also discuss some basic properties of elements of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group H.

scholarly journals On the associativity property of MPF over M16

2018 ◽  
Vol 59 ◽  
pp. 7-12
Author(s):  
Aleksejus Mihalkovich
Keyword(s):  
Power Function ◽  
Algebraic Structure ◽  
Interesting Problem ◽  
Algebraic Structures ◽  
Basic Properties ◽  
Matrix Power ◽  
One Way Function ◽  
Future Work

The objective of this paper is to find suitable non-commuting algebraic structure to be used as a platform structure in the so-called matrix power function (MPF). We think it is non-trivial and interesting problem could be useful for candidate one-way function (OWF) construction with application in cryptography. Since the cornerstone of OWF construction using non-commuting algebraic structures is the satisfiability of certain associativity conditions, we consider one of the possible choices, i.e. the group M16, explore its basic properties and construct templates to use in our future work. 

scholarly journals Neutrosophic Multigroups and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 95 ◽  
Author(s):  
Vakkas Uluçay ◽  
Memet Şahin
Keyword(s):  
Group Theory ◽  
Set Theory ◽  
Algebraic Structure ◽  
Group Structure ◽  
Algebraic Structures ◽  
Neutrosophic Sets ◽  
Basic Properties

In recent years, fuzzy multisets and neutrosophic sets have become a subject of great interest for researchers and have been widely applied to algebraic structures include groups, rings, fields and lattices. Neutrosophic multiset is a generalization of multisets and neutrosophic sets. In this paper, we proposed a algebraic structure on neutrosophic multisets is called neutrosophic multigroups which allow the truth-membership, indeterminacy-membership and falsity-membership sequence have a set of real values between zero and one. This new notation of group as a bridge among neutrosophic multiset theory, set theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroups and give its applications to group theory.

scholarly journals Pure Baer injective modules

1997 ◽  
Vol 20 (3) ◽  
pp. 529-538 ◽  
Author(s):  
Nada M. Al Thani
Keyword(s):  
Essential Extension ◽  
Injective Module ◽  
Algebraic Structures ◽  
Basic Properties ◽  
Injective Modules ◽  
Relative Complement

In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure Baer injective module. Some properties and some characterization of such modules are established. We also introduce two notions closely related to pure Baer injectivity; namely, the notions of a∑-pure Baer injective module and that of SSBI-ring. A ringRis an SSBI-ring if and only if every smisimpleR-module is pure Baer injective. To investigate such algebraic structures we had to define what we callp-essential extension modules, pure relative complement submodules, left pure hereditary rings and some other related notions. The basic properties of these concepts and their interrelationships are explored, and are further related to the notions of pure split modules.

A High Resolution Study of G.P. Zones in Aluminum - 4.2 at % Silver

1982 ◽  
Vol 40 ◽  
pp. 722-723 ◽  
Author(s):  
R. Gronsky
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
High Resolution ◽  
Computer Simulations ◽  
Structural Characteristics ◽  
X Ray Diffraction ◽  
X Ray ◽  
Transmission Electron ◽  
Resolution Study

The phenomenon of clustering in Al-Ag alloys has been extensively studied since the early work of Guinierl, wherein the pre-precipitation state was characterized as an assembly of spherical, ordered, silver-rich G.P. zones. Subsequent x-ray and TEM investigations yielded results in general agreement with this model. However, serious discrepancies were later revealed by the detailed x-ray diffraction - based computer simulations of Gragg and Cohen, i.e., the silver-rich clusters were instead octahedral in shape and fully disordered, atleast below 170°C. The object of the present investigation is to examine directly the structural characteristics of G.P. zones in Al-Ag by high resolution transmission electron microscopy.

SEM analysis of the change in the reaction rates with deposit structures in the copper-iron cementation system

1978 ◽  
Vol 36 (1) ◽  
pp. 456-457
Author(s):  
V. Annamalai ◽  
L.E. Murr
Keyword(s):  
Structural Characteristics ◽  
Secondary Electron Emission ◽  
Copper Deposit ◽  
Reaction Rates ◽  
Sem Analysis ◽  
Experimental Conditions ◽  
Major Drawback ◽  
Emission Mode ◽  
Scrap Iron ◽  
Image Position

Economical recovery of copper metal from leach liquors has been carried out by the simple process of cementing copper onto a suitable substrate metal, such as scrap-iron, since the 16th century. The process has, however, a major drawback of consuming more iron than stoichiometrically needed by the reaction.Therefore, many research groups started looking into the process more closely. Though it is accepted that the structural characteristics of the resultant copper deposit cause changes in reaction rates for various experimental conditions, not many systems have been systematically investigated. This paper examines the deposit structures and the kinetic data, and explains the correlations between them.A simple cementation cell along with rotating discs of pure iron (99.9%) were employed in this study to obtain the kinetic results The resultant copper deposits were studied in a Hitachi Perkin-Elmer HHS-2R scanning electron microscope operated at 25kV in the secondary electron emission mode.

Use of 2½ D imaging in analysis of NiTi martensite

1978 ◽  
Vol 36 (1) ◽  
pp. 610-611
Author(s):  
G. M. Michal
Keyword(s):  
Memory Effect ◽  
Monoclinic Phase ◽  
Reaction Path ◽  
Structural Characteristics ◽  
Salient Feature ◽  
Martensite Phase ◽  
Orientation Relationships ◽  
Low Symmetry ◽  
Hydride Phases ◽  
Unusual Shape

Several TEM investigations have attempted to correlate the structural characteristics to the unusual shape memory effect in NiTi, the consensus being the essence of the memory effect is ostensible manifest in the structure of NiTi transforming martensitic- ally from a B2 ordered lattice to a low temperature monoclinic phase. Commensurate with the low symmetry of the martensite phase, many variants may form from the B2 lattice explaining the very complex transformed microstructure. The microstructure may also be complicated by the enhanced formation of oxide or hydride phases and precipitation of intermetallic compounds by electron beam exposure. Variants are typically found in selfaccommodation groups with members of a group internally twinned and the twins themselves are often observed to be internally twinned. Often the most salient feature of a group of variants is their close clustering around a given orientation. Analysis of such orientation relationships may be a key to determining the nature of the reaction path that gives the transformation its apparently perfect reversibility.

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