Geometry on the lines of spine spaces

TL;DR

This paper demonstrates that the structure of lines and specific binary relations in spine spaces suffices to reconstruct the geometry of these spaces, highlighting the foundational role of coplanarity and pencils of lines.

Contribution

It establishes that lines, coplanarity, and pencils of lines form a minimal primitive system for describing spine spaces, and shows how to reconstruct the pencil geometry from these relations.

Findings

Lines and binary relations suffice to define spine space geometry.

Pencil of lines can be reconstructed from coplanarity and line relations.

Primitive notions are sufficient for the geometry of spine spaces.

Abstract

Spine spaces can be considered as fragments of a projective Grassmann space. We prove that the structure of lines together with binary coplanarity relation, as well as with binary relation of being in one pencil of lines, is a sufficient system of primitive notions for these geometries. It is also shown that, over a spine space, the geometry of pencils of lines can be reconstructed in terms of the two binary relations.