One Construction of an Affine Plane Over a Corps

Abstract

In this paper, based on several meanings and statements discussed in the literature, we intend constuction a affine plane about a of whatsoever corps (K,+,*). His points conceive as ordered pairs (α,β), where α and β are elements of corps (K,+,*). Whereas straight-line in corps, the conceptualize by equations of the type x*a+y*b=c, where a≠0K or b≠0K thevariables and coefficients are elements of that corps. To achieve this construction we prove some theorems which show that the incidence structure A=(P, L, I) connected to the corps K satisfies axioms A1, A2, A3 definition of affine plane. In all proofs rely on the sense of thecorps as his ring and properties derived from that definition.

Three Vertex and Parallelograms in the Affine Plane: Similarity and Addition Abelian Groups of Similarly <i>n</i>-Vertexes in the Desargues Affine Plane

“…, are similarity since from: 3 ), because parallelism in the affine plane is equivalence relation, which is described in [2], [3], [4].…”

“…Will generalize so its own meaning in the Euclidean case. With the help of parallelism [1], [2], [3] will give meaning of similarity and will see that have a generalization of the similarity of the figures in the Euclidean plane. By following the logic of additions of points in a line of Desargues affine plane submitted to [3], herewill show that analogously this meaning may also extend to the addition of similarity three-vertex in Desargues affine plane, moreover extend this concept for the similarity n-vertexes to the Desargues affine plane.…”

Skew-Field of Trace-Preserving Endomorphisms, of Translation Group in Affine Plane

“…Also great contributions in this direction have been made by, E.Artin in [1], H. S. M. Coxeter, in [3], Marcel Berger in [2], Robin Hartshrone in [5], etc. Even earlier, in my works ' [15], [18], [16], [4], [13], [14], [17], [20], [12], [19]' I have brought up quite a few interesting facts about the association of algebraic structures with affine planes and with 'Desargues affine planes', and vice versa.…”

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

“…Definition 5. [2,7,10] An affine plane complete with Desargues axiom (Proposition 1.3), is called Desargues affine plane.…”