Projective geometry on modular lattices

Ulrich Brehm, Marcus Greferath, Stefan E. Schmidt

Tutkimustuotos: Artikkeli kirjassa/konferenssijulkaisussaKirjan luku tai artikkeli muussa julkaisussa

Abstrakti

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.
AlkuperäiskieliEi tiedossa
OtsikkoHandbook of incidence geometry
KustantajaNorth-Holland
Sivut1115-1142
Sivumäärä28
ISBN (elektroninen)9780080533070
ISBN (painettu)978-0-444-88355-1
DOI - pysyväislinkit
TilaJulkaistu - 1995
OKM-julkaisutyyppiB2 Kirjan tai muun kokoomateoksen osa

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