A Topological Characterization of Hjelmslev’s Classical Geometries

Abstract

This article establishes a theoretical beginning for the study of topological Hjelmslev planes. Like the Drake-Jungnickel paper in this book, we are also interested in epimorphisms between incidence structures. But, rather than finite structures, we consider topological incidence structures; that is incidence structures whose point and line sets are topological spaces and where the joining of points and the intersecting of lines (where they exist) are continuous functions. In particular we look for factorizations called “solutions” of open continuous maps ∅: P → P′ where P and P′ are topological H-planes. The oldest and most elegant examples are constructed via homeogeneous coordinates over the topo-logical rings ℝ[x]/(xn). The maximal chain of ideals (0) ⊑ (xn−1)/(xn) ⊑ - - - - ⊑ (x)/(xn) generates a solution for this plane. It is our intention in these notes to determine all locally compact connected Pappian Hjelmslev Planes and to show that they all have solutions of the classical type. In fact, we prove that they are just the topological geometries over the rings K[x]/(xn), where K is the reals or complexes.