Four cross-ratio maps sets of Points and their Algebraic Structures in a line on Desargues Affine Plane

Abstract

This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatic and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, four types of cross-ratio maps sets of points, we discussed about for each of the 4-points of cross-ratio and we will examine the algebraic properties for each case. We are constructing four cross-ratio maps sets $\mathcal{R}^{A}_4=\left\{c_r(X,B;C,D) | \quad \forall X \in \ell^{OI} \right\}$, $\mathcal{R}^{B}_4=\left\{c_r(A,X;C,D) | \quad \forall X \in \ell^{OI} \right\}$, $\mathcal{R}^{C}_4=\left\{c_r(A,B;X,D) | \quad \forall X \in \ell^{OI} \right\}$ and $\mathcal{R}^{D}_4=\left\{c_r(A,B;C,X) | \quad \forall X \in \ell^{OI} \right\}$. We disuse and examine algebraic properties for each case, related to the actions of addition and multiplication of points in $\ell^{OI}$ line in Desargues affine planes, which are produced by these map sets.