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

Abstract

This paper considers dilations and translations of lines in the Desargues affine plane. A dilation of a line transforms each line into a parallel line whose length is a multiple of the length of the original line. In addition to the usual Playfair axiom for parallel lines in an affine plane, further conditions are given for distinct lines to be parallel in the Desargues affine plane. This paper introduces the dilation of parallel lines in a finite Desargues affine plane that is a bijection of the lines. Two main results are given in this paper, namely, each dilation in a finite Desarguesian plane is an isomorphism between skew fields constructed over isomorphic lines and each dilation in a finite Desarguesian plane occurs in a Pappian space.