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Full Description
This book is concerned mainly with the theory of flocks over skewfields. It begins with discussing what conditions would be required to find a possible way to extend flocks of hyperbolic quadrics and flocks of quadratic cones. This theory completely changes the idea of derivation of an affine plane that contains a derivable net.
This volume will give the necessary theory for the reader to understand how to construct examples and become researchers in the field. It shows how to construct four types of determinants, the (i,j)-determinants, which if never zero for the non-zero matrices of the spread will indicate that the first condition for existence of a spread then holds. If applicable, the left unwrapping principle, if this also is valid, will show that a left flock spread is constructed.
The book continues the presentation in Geometry of Derivations with Applications, Volume I, and a third volume, Geometry of Derivation, Volume III: Classification of Skewfield Flocks (2026) is also available, both from CRC Press. This is the sixth work in a longstanding series of books on combinatorial geometry by the author, including Subplane Covered Nets, Johnson (2000); Foundations of Translation Planes, Biliotti, Jha, and Johnson (2001); Handbook of Finite Translation Planes, Johnson, Jha, and Biliotti (2007); and Combinatorics of Spreads and Parallelisms, Johnson (2010), all published by CRC Press.
Like its predecessors, this book will primarily deal with connections to the theory of derivable nets and translation planes in both the finite and infinite cases. Translation planes over non-commutative skewfields have not traditionally had a significant representation in incidence geometry, and derivable nets over skewfields have only been marginally understood. Both are deeply examined in this volume, while ideas of non-commutative algebra are also described in detail, with all the necessary background given a geometric treatment.
Contents
Part 1: When Quasifibrations become Spreads 1. Quasifibrations 2. Unwrapping 3. Twisted Extensions 4. Semifield Planes from Cyclic Algebras 5. Proper Quasifibrations of Dimension 2 Part 2: Skewfield Flocks-A Window 6. The Main Points and Ideas Part 3: Foundations of Flock Theory 7. Building the Foundation 8. Generic and Non-Generic Flocks Part 4: Framework for Flock Theory 9. Setting up Flock and Spread Connections Part 5: Left σ-A Flocks and Spreads 10. Left σ - A-Hyperbolic Flocks 11. Left σ - A-Conical Flocks; 1st and 2nd Main Theorems 12. The Lower Left Form Theory 13. The 1st General Theorem of Flocks over Skewfields Part 6: Right τ - A∗ Flocks and Spreads 14. τ - A∗ Hyperbolic Flocks 15. Right τ - A∗ Hyperbolic 1st and 2nd Main Theorems 16. τ - A∗ Right Conical Flocks 17. The Right Upper Form Theory 18. Four "Easy" Problems Part 7: The Kaleidoscope of Derivable Nets 19. The Conical and Hyperbolic Isomorphism Questions Part 8: Apps of the Kaleidoscope 20. Resolution and Return-Flock Spreads 21. A Class of Linear 1 - Cc-Conical Flocks Part 9: Double Covers 22. The Left Generic Elation Double Nets Part 10: The Group of Conical Flock Spreads 23. Why Semifields? 24. Omnibus Theorem Part 11: Quaternion Division Ring Variations 25. 1-A Left Conical Spreads 26. Why Unwrapping? Part 12: Left Pseudo-Regulus-Inducing Homology Groups and Transposition 27. Inversing-Right Hyperbolic Spreads
