CHAPTER 21

Projective Geometry on Modular Lattices

Ulrich BREHM

Institut für Geometrie, Technische Universität Dresden, D-01062 Dresden, Germany

Marcus GREFERATH

FB Mathematik, Universität-GH-Duisburg, D-47048 Duisburg, Germany

Stefan E. SCHMIDT

FB Mathematik, Universität Mainz, D-55099 Mainz, Germany

Contents

1. Introduction

2. Preliminaries

3. Algebraic representation of modular lattices

3.1. Representation of complemented modular lattices

3.2. Representation of primary lattices

3.3. Partial algebraic representation of modular lattices with a homogeneous basis

4. Faltings' generalization of projective geometry

5. A lattice-geometric characterization of torsion free modules over left Ore domains

6. Projective lattice geometry

6.1. Concept

6.2. Desargues' postulate

6.3. Z-geometries

6.4. The U-property

6.5. A fundamental algebraization theorem

6.6. Point-uniform geometries

6.7. Reformulations of previous results

6.8. Point-irreducible geometries

6.9. Barbilian spaces in projective lattice geometries

7. Generalizations of the fundamental theorem

7.1. Representation of join and disjointness preserving mappings

7.2. Representation of lattice homomorphisms and lattice isomorphisms

References