On the Transitivity of Perspectivity in Continuous Geometries

Israel Halperin
1938 Transactions of the American Mathematical Society  
Introduction. The class of finite dimensional projective geometries has been extended to include non-finite dimensional ones by J. von Neumann's remarkable discovery of continuous geometries.]; In an axiomatic formulation of the geometry as an irreducible complemented modular lattice § the finiteness of the dimensionality is guaranteed by a chain condition. Von Neumann drops this chain condition and, retaining explicitly only two of its weak consequences, namely, completeness of the geometry
more » ... of the geometry and a certain continuity of the lattice operations, succeeds in establishing the existence of an essentially unique real-valued dimension function which may have either a discrete bounded range (the classical finite dimensional projective geometries) or a continuous bounded range (the new continuous geometries). In every case it is understood that the dimension function D(a) is to satisfy
doi:10.2307/1989895 fatcat:u456fyk53feqfpmxxhl5oar5za