Projective geometry on modular lattices

Ulrich Brehm, Marcus Greferath, Stefan E. Schmidt

Research output: Chapter in Book/Report/Conference proceedingOther chapter contribution

Abstract

This chapter focuses on projective geometry on modular lattices. Incidence and Order are basic concepts for a foundation of modern synthetic geometry. These concepts describe the relative location or containment of geometric objects and have led to different lines of geometry, an incidence-geometric and a lattice-theoretic one. Modularity is one of the fundamental properties of classical projective geometry. It makes projections into join-preserving mappings and yields perspectivities to be (interval) isomorphisms. It is therefore natural that order-theoretic generalizations of projective geometry are based on modular lattices and even more, the theory of modular lattices may be considered as a most general concept of projective geometry. In particular, the partially ordered set of all submodules of a module forms a (complete) modular lattice; even more general, any sublattice of the lattice of all normal subgroups of a group is a modular lattice. It considers that lattice-geometric approaches are complete geometrical structures whose geometrical objects form complete (modular) lattices.
Original languageUndefined/Unknown
Title of host publicationHandbook of incidence geometry
Pages1115-1142
Number of pages28
ISBN (Electronic)9780080533070
DOIs
Publication statusPublished - 1995

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