Abstract
The pointset E of an absolute plane \(({\bf E}, \mathcal{G}, \alpha, \equiv)\) can be provided with a binary operation "+" such that (E, +) becomes a loop and for each a \(\in\) E \ {o} the line [a] through o and a is a commutative subgroup of (E, +). Two elements a, b \(\in\) E \ {o} are called independent if [a] ∩ [b] = {o} and the absolute plane is called vectorspacelike if for any two independent elements we have E = [a] + [b] := {x + y | x \(\in\) [a], y \(\in\) [b]}. If \(({\bf E}, \mathcal{G}, \alpha, \equiv)\) is singular then (E, +) is a commutative group and \(({\bf E}, \mathcal{G}, \alpha, \equiv)\) is vectorspacelike iff \(({\bf E}, \mathcal{G}, \alpha, \equiv)\) is Euclidean. If \(({\bf E}, \mathcal{G}, \alpha, \equiv)\) is a hyperbolic plane then \(({\bf E}, \mathcal{G}, \alpha, \equiv)\) is vectorspacelike and in the continous case if a, b are independent, each point p has a unique representation as a quasilinear combination p = α · a + μ · b where α · a \(\in\) [a]and β · b \(\in\) [b] are points, α, β real numbers such that λ (o, λ · a) = |λ|· λ (o, a) and λ (o, μ · b) = |μ|. λ(o, b) and λ is the distance function.
Published Version
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