Linear Geometry and Algebra

TL;DR

Linear Geometry explores geometric properties based on lines, encoding information with a line relation, and develops algebraic structures from intuitive axioms using mostly synthetic proofs.

Contribution

This work provides a self-contained logical development of Linear Geometry, connecting geometric axioms to diverse algebraic structures and emphasizing purely geometric proofs.

Findings

Defines liners as sets with line relations

Classifies special classes of liners: regular, projective, affine, proaffine

Links geometric axioms to algebraic structures like magmas, loops, and fields

Abstract

Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So, Linear Geometry studies liners. Imposing some additional axioms on a liner, we obtain some special classes of liners: regular, projective, affine, proaffine, etc. Linear Geometry includes Affine and Projective Geometries and is a part of Incidence Geometry. The aim of this book is to present a self-contained logical development of Linear Geometry, starting with some intuitive acceptable geometric axioms and ending with algebraic structures that necessarily arise from studying the structure of geometric objects that satisfy those simple and intuitive geometric axioms. We shall meet many quite exotic algebraic structures that arise this way: magmas, loops, ternars, quasi-fields, alternative rings, procorps, profields, etc. The notion of area also belongs to Linear Geometry and can be defined and studied using only lines and their parallelity. We strongly prefer (synthetic) geometric proofs and use tools of analytic geometry only when no purely geometric proof is available. Liner Geometry has been developed by many great mathematicians since times of Antiquity (Thales, Euclides, Proclus, Pappus), through Renaissance (Descartes, Desargues), Early Modernity (Playfair, Gauss, Lobachevski, Bolyai, Poncelet, Steiner, M\"obius), Late Modernity Times (Steinitz, Klein, Hilbert, Moufang, Hessenberg, Jordan, Beltrami, Fano, Gallucci, Veblen, Wedderburn, Lenz, Barlotti) till our contempories (Hartshorne, Hall, Buekenhout, Gleason, Kantor, Doyen, Hubault, Dembowski, Klingenberg, Grundh\"ofer, M\"uller, Nagy).