Linear Geometry: flats, ranks, regularity, parallelity

TL;DR

This paper surveys fundamental concepts of Linear Geometry, focusing on properties related to flat hulls such as flats, exchange, rank, regularity, modularity, and parallelity.

Contribution

It provides a comprehensive overview of key notions and results in Linear Geometry centered around flat hulls, clarifying their interrelations.

Findings

Summarizes core properties of flats and ranks in Linear Geometry.

Explains the role of exchange, regularity, and modularity in geometric structures.

Highlights the importance of parallelity in the theory.

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.