Geometry of Derivation, Volume III

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, non-commutative skewfields, there are 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.