Advances in the Geometry of the Ratio of Linear Points in the Desargues Affine Plane Skew Field

TL;DR

This paper explores the geometric properties of ratios of points on lines in Desargues affine planes, establishing their algebraic structure as skew fields and analyzing their characteristics and mappings.

Contribution

It introduces a geometric framework for ratios of points in Desargues affine planes and constructs skew fields from these ratios, expanding the understanding of affine plane algebraic structures.

Findings

Ratios of points form skew fields on lines in Desargues affine planes.

Mappings of ratio point sets are bijections of lines.

Dyck polygons with co-linear ratio vertices have free group presentations.

Abstract

This paper introduces advances in the geometry of the ratio of either two or three points in a line in the Desargues affine plane, and we see this as a ratio of elements of skew field which are constructed over a line in Desargues affine plane. The results given here have a clean, geometric presentation based Desargues affine plan axiomatics and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. The results in this paper are: (1) study of properties for ratio of two and three points, in a line on Desargues affine plane. Also, we discuss the cases related to the "line-skew field" characteristic, when it is two and when it is different from two. (2) we have construct the maps for ratio points-set, for two and three points, and have prove that, this maps are bijections of the lines. (3) set of ratio points (for two and for three points) with addition and multiplication of points, forms a skew fields, for more, this skew fields are sub-skew fields of the 'line-skew field' on Desargues affine plane. (4) Every Dyck polygon containing co-linear ratio vertices in the Desargues affine plane has a free group presentation.