Geometry of Derivation, Volume III by Johnson, Norman L.;

Short description:

This book establishes the techniques, examples, and future directions of the specifics of flock theory over skewfields. It deals with connections to the theory of derivable nets and translation planes in both the finite and infinite cases. It is the seventh work in a series of research books on combinatorial geometry by the author.

Long description:

Geometry of Derivation, Volume III: Classification of Skewfield Flocks is the third book in a series of books on the topic. This book continues establishing the techniques, examples, and future directions of the specifics of flock theory over skewfields. Like its predecessors, it 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.

Since the work is valid for finite fields, infinite fields, left and right flocks over generalized hyperbolic quadrics, and generalized quadratic cones, there is a number of possibilities. The contribution of this volume is the main classification.

The book continues the presentation in Geometry of Derivations with Applications, Volume I, Johnson (2023), and Geometry of Derivation, Volume II: Theory of Skewfield Flocks (2026) is also available. This is the seventh 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.

Table of Contents:

Part 1:  The Classification of Flocks  1. The Classes of Flocks  2. General Theorems of Flocks  3. The Isomorphism Questions  Part 2: Multiple Replacement−Redux  4. Extension of Division Rings  5. Automorphism Groups of Division Rings  6. The Theorem of Andre’  7. Dickson Nearfield Planes  8. Ostrom’s Theorem  Part 3: Simultaneous Flock Spreads  9. Simultaneous Spreads of Type 2  Part 4: Semifields over Division Rings  10. Twisted T-Copies  11. General Skewfield Lifts to Semifields  12. Central Extensions of Degree 3, 4  13. Central Cyclic Extensions  Part 5: Lifting Skewsfields-Degree n  14. General Lifting  Part 6: Kantor-Pentilla and CJV σ – Flokki 15. Transform and CJV-Methods  16. Choices of Representation  Part 7: JPW-Hyperbolic Flocks  17. Idea of “Left-Inversion”  Part 8: Non-Linear Hyperbolic Flocks  18. Adjoining Inner Derivation Functions  19. Resolved Conical Flocks  20. The Isomorphism Questions  21. The Hyperbolic Isomorphism Question  Part 9: The Baer Flocks  22. Draxl's Theorem  23. Transposed Baer Flocks  Part 10: Anti-Isomorphic Flocks  24. The Hyperbolic Flock Square  Part 11: Elation Group Double Covers  25. The Three Spreads of a Double Cover  26. Skew-Desarguesian Spreads  27. Right Skew-Desarguesian Spreads  Part 12: Strings  28. Strings of Quasfibrations and Spreads  29. Corresponding Right “Flocks”  Part 13: Switch and Imposter Switch  30. Derivation of Flock Spreads  Part 14: Baer Groups over Skewfields  31. Point-Baer Subplanes of Planes  32. Baer Collineations in Translation Planes  33. Derived Spreads and Baer Groups  34. Deficiency One Flocks of Order p⁴   35. t_o-Interchange-Hyperbolic Spreads  36. t_o-Interchange-Conical Spreads  37. Left Inversing “Minus One”  38. Deficiency One  39. Hyperbolic Skew-Desarguesian Flocks  Part 15: Three Line Problem  40. Do Three Components define a Pseudo-Regulus?  41. Three Component-Three Point Construction  Part 16: The Flocks and Spreads  42. Anti-Isomorphic Flocks  43. Constructions-Generalized Lifted  44. 1-A Conical Spreads  45. Flocks from Lifted Types  46. The Open Types and New Directions