Vectorspacelike representation of absolute planes

The pointset E of an absolute plane can be provided with a binary operation "+" such that (E, +) becomes a loop and for each a E \ {o} the line a] through o and a is a commutative subgroup of (E, +). Two elements a, b 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 a], y b]}. If is singular then (E, +) is a commutative group and is vectorspacelike iff is Euclidean. If is a hyperbolic plane then 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 a]and β · b 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. This work was partially supported by the Research Project of MIUR (Italian Ministery of Education and University) “Geometria combinatoria e sue applicazioni” and by the research group GNSAGA of INDAM. Dedicated to Walter Benz on the occasion of his 75 ^th birthday, in friendship