Chain Geometry Determined by the Affine Group

Abstract

Abstract. Chain geometry associated with an affine group and with a linear group is studied. In particular, closely related to the respective chain geometries affine partial linear spaces and generalizations of sliced spaces are defined. The automorphisms of thus obtained structures are determined.Mathematics Subject Classification (2010). Primary 51B99, 51A45; Secondary 51B20.

The complement of a point subset in a projective space and a Grassmann space

“…As for which lines to remove from A our approach is not unique and there are different complements considered in the literature, e.g. in [13], where all the lines that meet W are deleted.…”

“…In [19] projective Grassmannians are successfully recovered from complements of their Grassmann substructures. The concept of two-hole slit space is introduced in [13]. It is a point-line space whose point set is the complement of the set of points of two fixed complementary subspaces, not hyperplanes, in a projective space and the line set is the set of all those lines which do not intersect any of these two subspaces.…”

“…every line of M has size at least 2 ind M (W) + 2. To recover improper lines we assume additionally that if U is a point and p is a line of M such that U / ∈ p and U, p span some strong subspace, then there is a proper maximal strong subspaces through U that meets p. (13) Our goal now is to prove an analogue of 3.1: Theorem 5.1. If M = P k (V ) is a Grassmann space and W is its point subset satisfying (2) and (13), then both M and W can be recovered in the complement D M (W).…”

“…Take any star or top X through U and any line p ⊂ X. By (13) there is a proper maximal strong subspace Y through U . So there is a proper point…”

“…a partial projective space. Observe that the assumption (12) is fulfilled in S(l) by (13). Hence, 4.1 can be applied, and thus all the remaining unrecovered lines of S(l) can be recovered now: the line l in particular.…”