Linear Geometry: flats, ranks, regularity, parallelity
Taras Banakh, Ivan Hetman, Alex Ravsky, Vlad Pshyk
TL;DR
This survey articulates a cohesive framework for Linear Geometry built on liners, focusing on flat hulls, ranks, and the interplay between Exchange, Regularity, and Parallelity. It systematically develops a hierarchy of liner classes (proaffine, affine, Playfair, Steiner, Hall, hyperaffine, Bolyai) and their first-order characterizations, linking geometric properties to algebraic structures like loops and magmas. Core results establish equivalences and implications among regularity, modularity, and parallelity, and examine the construction of planes, paralleled figures, and completions to parallelograms across diverse settings. The work not only consolidates known results but also highlights open questions and intricate connections to classical geometries (projective, affine, hyperbolic) and combinatorial designs (Steiner systems, unitals), with emphasis on the logical foundations via first-order axioms. The significance lies in providing a unified, axiomatized perspective on how line-based relations shape higher-dimensional incidence geometry and its algebraic encodings.
Abstract
Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results of Linear Geomery that depend on the flat hulls: flats, exchange, rank, regularity, modularity, and parallelity.