A Three Dimensional Modification of the Gaussian Number Field-Reference-Cited by-同舟云学术

A Three Dimensional Modification of the Gaussian Number Field

Published:2019-11-15 Issue:1 Volume:74 Page:63-76
ISSN:1210-3195
Container-title:Tatra Mountains Mathematical Publications
language:en
Short-container-title:

Author:

Haluška Ján¹,Jastrzębska Małgorzata²

Affiliation:

1. Mathematical Institute , Slovak Academy of Sciences , Košice Slovakia

2. Institute of Mathematics and Physics , Siedlce University of Natural Sciences and Humanities , Siedlce , Poland

Abstract

Abstract For vectors in E 3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications. Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ×ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.sciendo.com/pdf/10.2478/tmmp-2019-0020

Reference12 articles.

1. [1] ALPAY, A.—VAJIAC, A.—VAJIAC, M. B.: Gleason,s problem associated to a real ternary algebra and applications, Adv. Appl. Clifford Algebr. 28 (2018), no. 2, pp. 16. (Published online, paper no. 43)

2. [2] DROZD, YU. A.—KIRICHENKO, V. V.: Finite Dimensional Algebras. Springer-Verlag, Berlin, 1994.

3. [3] GREGOR, T.—HALUŠKA, J.: Lexicographical ordering and field operations in the complex plane, Stud. Mat. 41 (2014), no. 2, 123–133.

4. [4] HALUŠKA, J.: On fields inspired with a polar HSV-RGB theory of colour. In: Modern Real Analysis (J. Hejduk, S. Kowalczyk, R. Pawlak, M.Turowska, ed.),Wydawnictwo Universytetu Łódźkiego, Łódź 2015, pp. 69–88.

5. [5] HARKIN, A. A.—HARKIN, J. B.: Geometry of general complex numbers,Mathematics magazine, 77 (2004), no. 2, 118–129; https://www.researchgate.net/publication/265769569_Geometry_of_Generalized_complex_Numbers