Geometries of Degree n

Abstract

A geometry is of degree n (where n is any non-negative integer) if one and hence all of the 14 equivalent conditions of 4.4.1 and 4.4.4 are satisfied, and in particular if for any subspaces E, F the equation r(E ∨ F) + r(E ∧ F) = r(E) + r(F) holds provided that r(E ∧ F) ≥ n. The geometries of degree 0 are exactly the projective geometries. Among the geometries of degree 1 one finds the affine geometries and also the projective geometries (we remark that a geometry of degree n is trivially of degree n + 1). Within the geometries of degree 1 the projective geometries can be characterized by an axiom requiring that parallel lines must be equal, the affine geometries by an axiom requiring that for any line δ and any point p there exists a unique line parallel to δ containing p.