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Full 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.
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. to-Interchange-Hyperbolic Spreads 36. to-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
