Linear Geometry and Algebra (2506.14060v9)

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. 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).

• The paper introduces liner theory where a ternary relation defines lines and proves that any two distinct points uniquely determine a line.
• It rigorously develops concepts like flat sets, the exchange property, and rank, offering clear definitions to measure dimension and independence.
• The essay examines diverse liner types and explores regularity and parallelism axioms, establishing a unified framework for geometric structures.

Linear Geometry and Algebra: An Essay

This essay summarizes and analyzes the key concepts presented in Taras Banakh's book, Linear Geometry and Algebra. The book undertakes a systematic exploration of linear geometry, starting from fundamental geometric axioms and culminating in the algebraic structures that emerge from these axioms.

Liners and Lines

The foundational concept in the book is the liner, a set equipped with a ternary "line relation" $\mathsf{L}xyz$, signifying that point $y$ lies on the line determined by points $x$ and $z$. The line relation adheres to the Identity, Reflexivity, and Exchange axioms. A core object of paper is the $\mathsf{L}$-line, denoted $\overline{ab}$, which comprises all points $x$ satisfying $\mathsf{L}axb$. The book establishes fundamental properties of $\mathsf{L}$-lines, including the fact that a line is uniquely determined by any two distinct points it contains. The concepts of concurrent and colinear points, as well as closed, dense, and flat sets, are also introduced. Flat sets, crucial for further development, satisfy the property that the $\mathsf{L}$-line connecting any two points within a flat set is entirely contained within that set. Flat hulls and their properties, along with flat relations and functions between liners, are explored.

Exchange Property and Ranks

The book investigates the Exchange Property within liners, a property closely related to the concept of dimension in vector spaces. This property states that if a point $y$ lies in the flat hull of a flat set $A$ and a point $x$ outside of $A$, then $x$ must lie in the flat hull of $A$ and $y$. The book explores variations of the Exchange Property, such as the $\kappa$-Exchange Property, relevant to sets with cardinality less than $\kappa$. It establishes the equivalence between the $\kappa$-rankedness of a liner and its $\kappa$-Exchange Property. The rank and dimension of sets in liners, as well as the relative rank and codimension, are formally defined using the concept of flat hulls. These definitions facilitate the discussion of independent sets within liners and their crucial role in determining rank and dimension.

Regularity and Parallelism

The book introduces several Regularity Axioms that constrain the structure of the flat hull of a flat set with an additional point. Strongly regular liners, for instance, require the flat hull of a flat set $A$ and a point $b$ outside of $A$ to be the union of all lines connecting $b$ with points in $A$. The book then explores parallelism in liners, introducing a spectrum of Parallelity Postulates and Axioms, including those of Proclus, Playfair, Bolyai, and Lobachevsky. These axioms and postulates differentiate liners based on the number of lines parallel to a given line through a point not on the given line, within a given plane. The connections between various Parallelity Axioms and Regularity Axioms are explored, revealing that projective liners, for instance, are equivalent to strongly regular liners. The book also introduces the notion of balanced liners, where flat sets of the same rank have the same cardinality. Balancedness plays a significant role in analyzing the structure of finite liners.

Projective, Affine, and Other Liners

The book examines specific types of liners, including projective, proaffine, affine, hyperaffine, hyperbolic, and injective liners, each defined by specific Parallelity Axioms. Projective liners, for example, are those where any two lines within a plane intersect. The properties of these liners are examined in detail, including characterizations of their $L$-lines and their balancedness properties. The book constructs the spread completion of a $3$-ranked liner, adding a "horizon" of directions represented by spreads of parallel lines. The relationship between the original liner and its spread completion is thoroughly investigated, with a focus on completely regular liners whose spread completions are projective.

Implications and Future Directions

The systematic approach of Linear Geometry and Algebra highlights the deep connections between seemingly disparate geometric and algebraic concepts. The formal definitions of liners, lines, rank, dimension, and various types of liners, alongside the introduction of related algebraic structures such as magmas, loops, and ternary rings, provide a rich framework for exploring the foundations of geometry and its interplay with algebra. The many examples in the book demonstrate the variety of liners, especially finite liners, and the algebraic subtleties they embody. The strong numerical results presented, such as those related to the cardinality of liners with specific properties, offer concrete examples for further investigation. The open problems posed at the end of the book point to areas of ongoing research. The paper of proaffine liners, which generalize both affine and projective liners, could lead to a unified theory encompassing these seemingly different geometric frameworks. The classification of liners based on their properties, including the density and automorphism group, suggests further avenues for exploring the structural diversity of liners.

Conclusion

This book provides a rigorous framework for understanding linear geometry. The detailed exploration of liners, their properties, and the corresponding algebraic structures provides valuable tools for researchers in geometry and related fields. The book's clarity and rigor make it a significant contribution to the mathematical literature.