ABSTRACT
In this chapter we develop the theory of the finite analogs of Euclidean and projective geometry. An incidence structure consists of two sets, a set P of points and a set L of lines, together with a binary relation of incidence between elements of P and elements of L. If a point p is incident with a line l, one says “p lies on l” or “l contains p.” With any line one can associate a subset of P , namely the set of all points that lie on the given line. We shall never wish to discuss geometrical situations where a line can contain no points or where two different lines can have the same set of points, so we can in fact define lines to be nonempty sets of points. We define a geometry to consist of a set P of objects called points and a set L of nonempty subsets of P called lines that satisfy the two axioms (A1) and (A2):
(A1) given any two points, there is one and only one line that contains them both;
(A2) there is a set of four points, no three of which belong to one common line.
