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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

malashkevichvb@volgatech.net
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

malashkevichia@volgatech.net

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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