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

Orgest Zaka

TL;DR

This paper studies cross-ratio maps on a line in a Desargues affine plane, addressing how four-point cross-ratios induce algebraic structures on the line. It constructs four map-sets $R^A_4$, $R^B_4$, $R^C_4$, $R^D_4$ and analyzes their additive and multiplicative properties, revealing Möbius-point structure in $R^A_4$ and group structures in the other sets. Explicit neutral elements and inverses are identified for each set (e.g., additive zeros at $mu(C)$, $f_B(C)$, $f_C(A)$, $f_D(B)$, and multiplicative units at $mu(B)$, $f_B(A)$, $f_C(D)$, $f_D(C)$). Together, these results illuminate how cross-ratio maps enact skew-field-like operations on a Desargues line, contributing to the algebraic modeling of Desargues affine geometry.

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.

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

TL;DR

This paper studies cross-ratio maps on a line in a Desargues affine plane, addressing how four-point cross-ratios induce algebraic structures on the line. It constructs four map-sets , , , and analyzes their additive and multiplicative properties, revealing Möbius-point structure in and group structures in the other sets. Explicit neutral elements and inverses are identified for each set (e.g., additive zeros at , , , , and multiplicative units at , , , ). Together, these results illuminate how cross-ratio maps enact skew-field-like operations on a Desargues line, contributing to the algebraic modeling of Desargues affine geometry.

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 , , and . We disuse and examine algebraic properties for each case, related to the actions of addition and multiplication of points in line in Desargues affine planes, which are produced by these map sets.

Paper Structure

This paper contains 7 sections, 12 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Four 'cross-ratio map sets' of points in a line of Desargues affine plane and their algebraic properties.
  3. Algebraic properties for the 'cross-ratio map sets' $\mathcal{R}^{A}_4$, set of Mobiüs points
  4. Algebraic properties for the 'cross-ratio map sets' $\mathcal{R}^{B}_4$-set of points
  5. Algebraic properties for the 'cross-ratio map sets' $\mathcal{R}^{C}_4$-set of points
  6. Algebraic properties for the 'cross-ratio map sets' $\mathcal{R}^{D}_4$-set of points
  7. Declarations

Key Result

Theorem 1

Let's have three different-fixed points $B, C, D \in \ell^{OI}$ and this points are different for point $O$, in $\ell^{OI}-$line. The Mobiüs transforms $\mu(\cdot)$ from $\mathcal{R}_4$, with addition of points, fulfills the properties: (1) is Associative, (2) is commutative and, (3) it has a neutra

Figures (4)

  • Figure 1: Desargues Axioms: (a) For parallel lines $\ell^{AA'} \parallel \ell^{BB'} \parallel \ell^{CC'}$; (b) For lines which are cutting in a single point $P$, $\ell^{AA'} \cap \ell^{BB'} \cap \ell^{CC'}=P$.
  • Figure 2: (a) Addition of points in a line in affine plane, (b) Multiplication of points in a line in affine plane
  • Figure 3: Ilustrate the Ratio-Point, of 2-Points in a line of Desargues affine plane $R=r(A:B)=B^{-1}A$.
  • Figure 4: Ratio of 3-Points in a line of Desargues affine plane $R=r(A,B;C)$.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 7 more