Generalizations of Einstein Numbers by Adding New Dimensions to Domains Preserving Commutativity and Associativity
Let c > 0. Let R∞ = R ∪ {∞} be the projective real line with one additional compactification element, denoted as ∞. Under the set of all real Einstein numbers we understand the system E∞
(–c, c]
= ((–c, c] = ϕ(R∞), ⊕), where ϕ : R∞ → (–c, c] is a bijective function given as follows: ϕ(∞) = c and ϕ(v) = c tanh(v) for every v
∈ R. In this paper, we generalize Einstein numbers such that we sequentially add new dimensions (coordinates) and this way we extend the domains where we define the extended addition. The extended generalized additions in the same time have to be commutative and associative of the newly
defined generalized operation of addition in the process of adding of new dimensions. For the dimension 2, we also found a bounded semi-field. We bring examples.
Keywords: ASSOCIATIVITY; COMMUTATIVITY; EINSTEIN NUMBERS; GENERALIZED EINSTEIN ADDITION; HYPERBOLIC ADDITION
Document Type: Research Article
Publication date: 01 June 2015
- Journal of Advanced Mathematics and Applications (JAMA) publishes peer-reviewed research papers in mathematics in general, covering pure mathematics and applied mathematics as well as the applications of mathematics in chemistry, physics, engineering, biological sciences/health sciences, brain science, computer and information sciences, geosciences, nanoscience, nanotechnology, social sciences, finance and other related fields.
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