Locally generalized projective lattices satisfying a bundle theorem

Abstract

The question of embedding locally generalized projective geometries in generalized projective geometries has been considered by a number of people, and in various settings ([1], [2], [6]-[14]). The most popular cadre seems to have been the lattice-theoretic one. Traditionally ({-1], [2], [10]-[14]), the assumption of rank strictly greater than 4 has been made in embedding theorems of this type, and then a natural geometric constructive argument is possible. It is known that in the rank-4 case the same results cannot be obtained; counterexamples appear in [8] and [10]. Under additional assumptions, the rank-4 case has been handled by Herzer [6]-[8] and Kahn [9] more recently and in quite different ways. In [7] Herzer supposes a lattice situation allowing lines (rank-2 elements) with a single point (rank-1 element). He supposes, however, that all 1-point lines on any given point are contained in a single plane on the point. Under this condition a rank-4 semimodular locally projective (of order >-4) lattice is shown to be embeddable in a modular one precisely when a certain configuration holds throughout the lattice. It is of interest to note that central collineations are introduced in the proof.Kahn [9] supposes a weaker condition on the 1-point lines, but this leads, it seems, to a more complex proof in which at one point the finite and infinite cases must be handled differently. However, the configuration he shows to be necessary and sufficient for embeddability is much simpler (in a geometrical sense) than that considered by Herzer. He also assumes that the lattice is locally projective.In this article, we present an embedding theorem which unifies the rank-4 and rank >4 cases. We prove that a particular situation is necessary and sufficient in very general circumstances. We suppose: rank >-4, locally generalized projective, outside each plane (rank-3 element) there be at least one point, and outside each rank-4 element there be at least one point unless the lattice itself has rank-4. Thus elements of rank >`2 may contain a single point. We note that the conditions used by Herzer and Kahn imply that there is a point outside each plane.The proof of the embedding is geometrically natural and very similar to that given by Kantor [10] and also that used in [1] and [2]. We divide our

Coplanarity of lines in projective and polar Grassmann spaces

“…Most of the time, e.g. in [1,15,18,34], the Bundle Theorem is the necessary condition for the construction of the bundle space in locally projective spaces. We do not follow exactly this path.…”

“…We try to imitate the approach of [18] and other papers, e.g. [1,15,34], where the construction of a bundle space is used to recover various geometries embeddable into a projective space. However, we do not use it to settle the problem of projective embeddability.…”

Linear spaces with many projective planes

“…In many papers the question is discussed which linear space (M, M) can be embedded into a projective space (cf. [1,2,3,4,5,9,10,12]). The usual method for projective embedding is the construction of the point set P of the projective space (P , L) by line bundles of the given linear space (M, M).…”