DYCK FREE GROUP PRESENTATION OF POLYGON CYCLES IN THE RATIO OF COLLINEAR POINTS IN THE DESARGUES AFFINE PLANE

This paper introduces Dyck free groups that are geometric realizations of polygon cycles, the ratio of either two or three collinear points in the Desargues affine plane. A biproduct of this work is a concise presentation of Desargues affine planar cycles as a free group. In this paper, we include (1) the study of properties for the ratio of two and three points in a line in the 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 constructed the maps for ratio points-set for two and three points and have proved that these maps are bijections of the lines. (3) We observe that the set of ratio points (for two and for three points) with addition and multiplication of points forms a skew field and these skew fields are sub-skew fields of the “line-skew field” in a Desargues affine plane. (4) We also observe that every Dyck polygon cycle in the Desargues affine plane is the realization of a Dyck path cycle. (5) We prove that every Dyck polygon containing collinear ratio vertices has a free group presentation.