General Finite Geometries

In recent years the subject of miniature or finite geometries has aroused some interest. Most studies to date have been concerned with the study of finite projective and affine (Euclidean) geometries, Many teachers have found this material simple enough to present to students studying geometry and algebra in high school. This paper proposes to describe axiomatically a large class of finite systems called finite Bolyai-Lobachevsky geometries. The approach will be geometric instead of algebraic, although to pursue the subject in depth algebraic techniques are needed. The Bolyai-Lobachevsky geometries can be presented in somewhat the same manner as the studies of finite projective geometries.