Linear Geometry: flats, ranks, regularity, parallelity

Linear geometry describes geometric properties that depend on the fundamental concept of a line. In this article, we survey the main concepts and results of linear geometry that depend on the flat hulls: flats, exchange, rank, regularity, modularity, and parallelity.The article is the first in a series of reviews of linear geometry, the branch of geometry that studies geometric properties that depend only on lines, scheduled for publication in the journal Matematychni Studii. Linear geometry emerged as a separate branch of geometry in the 20th century. We present the main ideas and results of linear geometry using Bourbaki’s approach to mathematics as a science that studies mathematical structures, in particular, such a fundamental mathematical structure of linear geometry as the structure of a liner.