On n-uniform Hjelmslev planes

We define 1-uniform and strongly 1-uniform Hjelmslev planes (H-planes) to be the ordinary affine and projective planes. An n-uniform H-plane (n>1) is an H-plane whose point neighborhoods all are (n−1)-uniform affine H-planes. A strongly n-uniform H-plane (n>1) is an n-uniform projective H-plane which collapses to a strongly (n−1)-uniform H-plane upon identifying maximally connected points (points joined by t lines). All uniform projective H-planes are strongly n-uniform with n=1 or n=2. It is proved that all Desarguesian projective H-planes are strongly n-uniform. Many nice intersection properties are given for n-uniform H-planes; strongly n-uniform H-planes satisfy a strong intersection property called “property A.” It is proved that an n-uniform projective H-plane π is strongly n-uniform if and only if π satisfies property A, and also if and only if π^*, the dual of π, is n-uniform.