Lexicographical ordering and field operations in the complex plane 

Author 
gregor@saske.sk, jhaluska@saske.sk
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 socalled 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), 177–194.
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 LT50168, Jasalcio g.7, Lithuania. 4. E.M. Vechtomov, A.V. Cheraneva, Semifields and their properties, Jour. of Math. Sciences, 163 (2006), ¹6, translated from Fundamentalnaya i prikladnaya matematika, 14 (2008), ¹5, 3–54. (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 
123133

Volume 
41

Issue 
2

Year 
2014

Journal 
Matematychni Studii

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