Introduction of measures for segments and angles in a general absolute plane (Q2468001)

The authors deal with the absolute plane \((E,{\mathcal L},\equiv,\alpha)\) where \(E\), resp. \({\mathcal L}\), denotes the set of points, resp. lines, \(\equiv\) the congruence relation between pairs of points and \(\alpha\) the order structure. They make no claim concerning continuity or the Archimedian axiom. They collect properties if the absolute plane which will be used in order to introduce a measure for segments and measure for angles. An ordered commutative group \((W, +, <)\) such that \((W, +)\) is a subgroup of the corresponding \(K\)-loop \((E, +)\) of the absolute plane and a cyclic ordered commutative group \((E_1,\cdot,\zeta)\) where \((E_1,\cdot)\) is isomorphic to a rotation group fixing point are defined. Properties of the measure groups mentioned above are used for giving of absolute value and distance. The next parts of the present paper are dealing with induced separation, cyclic order on circles and measure of angles.