Lexicographical ordering and field operations in the complex plane

T. Gregor; J. Haluska

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 $\mathbb{C}$.

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 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, 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.

	Lexicographical ordering and field operations in the complex plane
Author	T. Gregor, J. Haluska 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 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), 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 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, 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	123-133
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
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Table of content of issue	html