Lexicographical ordering and field operations in the complex plane

Author
T. Gregor, J. Haluska
Mathematical Institute, Slovak Academy of Sciences, Kosice
Abstract
Let $K \in \mathbb{N}, K \geq 3$. Let $\mathbb{C}_K = \mathbb{R}_0^K$ be the Cartesian product of $K$ copies of $\mathbb{R}_0$, where $\mathbb{R}_0$ denotes the set of all nonnegative real numbers. We equip this set with arithmetic operations and show that under the condition of the so-called Cancelation Law, the space $\mathbb{C}_K$ is arithmetically isomorphic with the standard field $\mathbb{C}$ of complex numbers. Distinct $K \in \mathbb{N}$ determine distinct lexicographical orderings on $\mathbb{C}$.
Keywords
lexicographical order; field operations; complex plane; isomorphism
Reference
1. E. Haluskova, Two element direct limit classes of monounary algebras, Math. Slovaca, 52 (2002), 177194.

2. V. Lenski, Generation of multi polarity electromagnetic energy, US patent from 15.11.2006, WO 2008/060342, PCT/US2007/01707.

3. N. Kutka, S. Goceikis, S. Adamovicius, Multipolarity radio telescope and radio transmission systems, Materials of the Multi polarity laboratory of cryptography, http://www.dkl.lt, Kaunas LT-50168, Jasalcio g.7, Lithuania.

4. E.M. Vechtomov, A.V. Cheraneva, Semifields and their properties, Jour. of Math. Sciences, 163 (2006), 6, translated from Fundamentalnaya i prikladnaya matematika, 14 (2008), 5, 354. (in Russian)

5. J.S. Golan, Some recent applications of semiring theory, Int. Conf. on Algebra in memory of Kostia Beidar, National Cheng Kung University, Taiwan, 2005, 18 p.
Pages
123-133
Volume
41
Issue
2
Year
2014
Journal
Matematychni Studii
Full text of paper
pdf
Table of content of issue