Extended parallelity in spine spaces and its geometry

Abstract

The notion of extended parallelity is introduced in an arbitrary spine space, and rudimentary properties of the obtained geometry are presented. The extended parallelity is used in the development of the theory of spine spaces. Also, the horizon and dilatation group relative to this parallelity are examined. (2002): 51A15, 51A45. Mathematics Subject Classification

Geometry on the lines of spine spaces

“…A spine space is a fragment of a Grassmann space chosen so that it consists of subspaces of V which meet a fixed subspace in a specified way. The concept of spine spaces was introduced in [10] and developed in [11], [12], [14], [13].…”

“…From that point onwards the reasoning is unified and we get that π as well as ρ is a sufficient primitive notion for the geometry a spine space M. Generally, as in the case of Grassmann spaces, the key role play maximal cliques of π and ρ, as well as maximal strong subspaces containing them. The structure of strong subspaces in spine spaces is much more complex than in Grassmann spaces, but pretty well known if we take a look into [12], [11], [13], and [14]. The major difference is that we have to deal with three types of lines and four types of strong subspaces as stars and tops can be projective or semiaffine (in particular affine) spaces.…”