Chapter 10 - Some Classes of Rank 2 Geometries

Abstract

The focus of this chapter is on classes of rank 2 geometries. A rank 2 geometry S is a {0, 1}-geometry. The elements of type 0 will be called the points while the elements of type 1 will be called lines. A rank 2 geometry S is called a partial linear space, if each point is on at least 2 lines, if all lines have at least two points and if any two distinct points in P are incident with at most one line, or equivalently, if any two distinct lines are incident with at most one point. It is also called as a semi-linear space. Lines incident with only 2 points, are called thin lines. If all lines are thin lines, then S is called a thin partial linear space. If all lines are incident with at least 3 points and if every point is incident with at least 3 lines, the partial linear space is called thick. Two points are said to be collinear if they are incident with a common line. Dually, two lines are said to be concurrent if they are incident with a common point.