Projective Geometry and Algebraic Structures

Projective Geometry and Algebraic Structures focuses on the relationship of geometry and algebra, including affine and projective planes, isomorphism, and system of real numbers.

The book first elaborates on euclidean, projective, and affine planes, including axioms for a projective plane, algebraic incidence bases, and self-dual axioms. The text then ponders on affine and projective planes, theorems of Desargues and Pappus, and coordination. Topics include algebraic systems and incidence bases, coordinatization theorem, finite projective planes, coordinates, deletion subgeometries, imbedding theorem, and isomorphism.

The publication examines projectivities, harmonic quadruples, real projective plane, and projective spaces. Discussions focus on subspaces and dimension, intervals and complements, dual spaces, axioms for a projective space, ordered fields, completeness and the real numbers, real projective plane, and harmonic quadruples.

The manuscript is a dependable reference for students and researchers interested in projective planes, system of real numbers, isomorphism, and subspaces and dimensions.

Contenu

Preface
Acknowledgments
Chapter 1 Introduction
1.1 Euclidean Planes
1.2 Incidence Bases
1.3 Set Theory
Chapter 2 Affine Planes
2.1 Axioms for an Affine Plane
2.2 Examples
Chapter 3 Projective Planes
3.1 Axioms for a Projective Plane
3.2 Examples
3.3 Algebraic Incidence Bases
3.4 Self-Dual Axioms
Chapter 4 Affine And Projective Planes
4.1 Isomorphism
4.2 Deletion Subgeometries
4.3 The Imbedding Theorem
Chapter 5 Theorems of Desargues and Pappus
5.1 Configurations
5.2 Theorem of Desargues
5.3 Theorem of Pappus
Chapter 6 Coordinatization
6.1 Coordinates
6.2 Addition
6.3 Multiplication
6.4 Algebraic Systems and Incidence Bases
6.5 The Coordinatization Theorem
6.6 Finite Projective Planes
Chapter 7 Projectivities
7.1 Perspectivities and Projectivities
7.2 Some Classical Theorems
7.3 A Nonpappian Example
Chapter 8 Harmonic Quadruples
8.1 Fano Axiom
8.2 Harmonic Quadruples
Chapter 9 The Real Projective Plane
9.1 Separation
9.2 Ordered Fields
9.3 Completeness and the Real Numbers
9.4 Separation for Basis 3.5
9.5 The Real Projective Plane
9.6 Euclidean Planes
Chapter 10 Projective Spaces-Part 1
10.1 Axioms for a Projective Space
10.2 Examples
Chapter 11 Projective Spaces-Part 2
11.1 Subspaces and Dimension
11.2 Intervals and Complements
11.3 Dual Spaces
Appendix A Hilbert's Axioms for a Euclidean Plane
Group I. Axioms of Connection
Group II. Axioms of Order
Group III. Axiom of Parallels
Group IV. Axioms of Congruence
Group V. Axiom of Continuity
Appendix B Division Rings
Appendix C Quaternions
References
Index of Special Symbols
Subject Index