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Cybernetics and programming
Reference:
Malashkevich V.B., Malashkevich I.A. —
Elements of algebra of triplexes in idempotent bases
// Cybernetics and programming.
– 2016. – № 1.
– P. 1 - 228.
DOI: 10.7256/2306-4196.2016.1.17583 URL: https://en.nbpublish.com/library_read_article.php?id=17583
Elements of algebra of triplexes in idempotent bases
Malashkevich Vasilii Borisovich
PhD in Technical Science
Associate Professor, Department of Computer Science and Engineering, Volga State University of Technology
424038, Russia, respublika Marii El, g. Ioshkar-Ola, Leninskii pr., 14, kv. 121
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malashkevichvb@volgatech.net
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Malashkevich Irina Ardalionovna
Associate Professor, Department of Information and Computer Systems, Volga State University of Technology
656035, Russia, Altaiskii krai, g. Barnaul, pr. Lenina, 61
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malashkevichia@volgatech.net
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DOI: 10.7256/2306-4196.2016.1.17583
Review date:
14-01-2016
Publish date:
11-02-2016
Abstract. The subject of study in algebra is a ternary (three-dimensional) hypercomplex numbers (triplexes). Since the time of Hamilton (1983) algebras of hypercomplex numbers attracted the attention of researchers. The largest number of papers in this area is dedicated to quaternion algebra and bicomplex numbers, as well as its applications to the solution of various problems of science and technology. Algebra of ternary (three-dimensional) hypercomplex numbers is less studied. However, it is undoubtedly promising in solving problems related to processing of point objects and fields in three-dimensional Euclidean space. The main objective of the article is forming a basis of idempotent algebra of three-hypercomplex numbers. Idempotent bases are typical for commutative multiplicative algebras without division. Such bases provide a simple definition and way of studying mathematical constructions of hypercomplex numbers as well as a significant increase in computational efficiency. The paper presents all possible unit vectors of potential idempotent bases of triplex numbers. The authors highlight two idempotent bases providing not excessive presentation of triplexes. The main attention is given the study of one of these bases with complex unit vectors. The paper shows, that idempotent triplexes basis allows formulating the definition of arithmetic operations and triplex argument functions in terms of the well-studied algebra of real and complex numbers. At the same time mentioned basis provides a high computational efficiency for calculating values of these operations and functions.
Keywords:
hypercomplex numbers, commutative hypercomplex algebra, zero divisor, idempotent basis, triplex, Algebra without division, triplex algebra, conjugation, triplex function of the argument, triplex ring
This article written in Russian. You can find full text of article in Russian
here
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