The Construction of a Corp in the Set of Points in a Line of Desargues Affine Plane

Abstract

In the article [1], we show that the set of points on a line, in the affine Desargues plans, connected with addition forms an Abelian group. In this article, we will define multiplication of points on a line in the affine Desargues plans. We will show that this set forms a multiplicative group. And we will show that every straight line of Desargues affine plans, along with both addition and multiplication operations, forms the corp (skew-field).1. Introduction, Desargues affine plane, commutative group (OI, +) Definition 1.

Isomorphic-Dilations of the skew-fields constructed over parallel lines in the Desargues affine plane

“…In our case, the role of zero is the fixed point O. From the addition algorithm (Algorithm 1 in [21] and [6]) we have:…”

“…O", A", C" ∈ ℓ 2 = δ (ℓ 1 ) and B" ∈ δ (OB) . Also by definition and dilation properties (see [6]), we have:…”

“…Now calculate the addition of points A ′ = ϕ (A) and C ′ = ϕ (C). From the addition algorithm (Algorithm 1 in [21] and [6]), we have:…”

“…The foundations for the study of the connections between axiomatic geometry and algebraic structures were set forth by Hilbert [7], recently elaborated and extended in terms of the algebra of affine planes in, for example, [8], [3, §IX.3, p. 574](affine transformation of the plane transforms a lattice into a lattice) and in the Desargues affine plane (see, e.g., [20,21,6,18,19,4,17]). Properties of geometric structures such as adjacency and proximity [2, §III.…”

“…In previous work [21], we have shown how we can transform a line in the Desargues affine plane into an additive Group of its points. We have also shown [6] how to construct a skew-field with a set of points on a line in the Desargues affine plane. In addition, for a line of in any Desargues affine plane, we construct a skew-field with the points of this line, by appropriately defined addition and multiplication of points in a line.…”