33531320
The complement of a point subset in a projective space and a Grassmann space
Abstract In a projective space we fix some set of points, a horizon, and investigate the complement of that horizon. We prove, under some assumptions on the size of lines, that the ambient projective space, together with its horizon, both can be recovered in that complement. Then we apply this result to show something similar for Grassmann spaces.
119424033
Chain Geometry Determined by the Affine Group
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.
118592297
Affine polar spaces
Affine polar spaces are polar spaces from which a hyperplane (that is a proper subspace meeting every line of the space) has been removed. These spaces are of interest as they constitute quite natural examples of ‘locally polar spaces’. A characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine. Moreover, the affine polar spaces are fully classified in the sense that all hyperplanes of the fully classified polar spaces (of rank at least 3) are determined.
118425465
Affine polar spaces derived from polar spaces and Grassmann structures defined on them
We prove that an affine polar space in the meaning of Cohen and Shult can be recovered from one of the three adjacency relations on a Grassmann structure over it. The result directly generalizes the results of our previous work where we use an affine space over a vector space equipped with a nondegenerate reflexive form as a starting point to the Cohen-Shult affine polar spaces.
121799048
Subspaces and Parallelity in semiaffine partial linear spaces
In the paper we characterize subspaces and strong subspaces of a semiaffine partial linear space and consider definability of projective and semiaffine planes, affine lines and parallelity in terms of projective lines. We also give some construction of a wide class of semiaffine partial linear spaces.
120427726
On a construction of affine Grassmannians and spine spaces
Abstract. In this paper we introduce the notion of spine space, generalizing the notion of affine grassmannians. We describe the set of strong subspaces of a spine space (Thm. 2.6, 2.7), construct the horizon of a spine space, and study the automorphisms of a spine space.
120456706
Automorphisms of spine spaces
Automorphisms of non-trivial spine spaces are either, type-preserving, or type-exchanging, that is, they map stars onto stars or exchange them with tops. In the first case they are given by semi-linear bijections, in the latter, by sesqui-linear forms.
119880607
Geometry of the structure of linear complements
Abstract.Two geometries can be considered in the structure of linear complements: an 
affine spine space and an affine space. An affine spine space arises from a space of
pencils. In terms of this geometry an affine partial line space may be defined. It is
extensible to the affine space. Automorphisms of the affine spine space are
automorphisms of appropriate affine space.
17847164
Affine Spaces within Projective Spaces
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.
119481275
Complements of Grassmann substructures in projective Grassmannians
We prove that a projective Grassmannian can be recovered from the complement of one of its Grassmann substructures. Even more, the underlying projective space with the interval of its distinguished subspaces can be recovered.
121726092
Ein Axiomensystem für partielle affine Räume
A partial linear space with parallelism is called partial affine space if it is embeddable in an affine space with the same pointset preserving the parallelism. These partial affine spaces will be characterized by a system of three axioms for partial linear spaces with parallelism.
116834517
Chapter 12 – Point-Line Spaces Related to Buildings
Publisher Summary 
This chapter focuses on point-line spaces related to buildings. The definition of space is very general in that any collection of nonempty, nonsingleton subsets of any set can be the set of lines of a space. A stricter class of spaces is formed by the partial linear spaces, i.e. spaces in which no two different points are on two different lines. For two distinct collinear points x, y of such a space, xy is written to denote the line containing them. A hyperplane of a space S is a proper subspace H of S, such that any line of S has at least one point in H. Thus, if S is a projective space, its hyperplanes are the usual maximal proper subspaces. A polar space is a space in which, for every point x, the subset x⊥ is either a hyperplane or the whole point set. Because of the definition, the polar spaces are gamma spaces of diameter at most 2. Singular spaces are degenerate examples of polar spaces. A polar space is said to be of rank n if its singular rank is n − 1.
118845849
GEOMETRY OF POLAR GRASSMANN SPACES
The Grassmann space of fc-subspaces of a polar space is defined and its geometry is examined. In particular, its cliques, subspaces and automorphisms are characterized. An analogue of Chow's theorem for the Grassmann space of fc-subspaces of a polar spaces is proved.
707088
Grassmannians of Classical Buildings
Linear Algebra and Projective Geometry Buildings and Grassmannians Classical Grassmannians Polar and Half-Spin Grassmannians.
27771502
Correlations of spaces of pencils
Abstract The paper introduces an axiomatic system of a conjugacy in partial linear spaces, and provides its analytical characterization in spaces of pencils. A correlation of a space of pencils is defined and it is shown to correspond to a polarity of the underlying projective space, i.e. to a reflexive sesqui-linear form, or also to an involutory collineation, i.e. to an injective semi-linear map, in the self-dual case. A geometric characterization of segment subspaces in spaces of pencils is also provided.