Combinatorics of Spreads and Parallelisms

Combinatorics of Spreads and Parallelisms covers all known finite and infinite parallelisms as well as the planes comprising them. It also presents a complete analysis of general spreads and partitions of vector spaces that provide groups enabling the construction of subgeometry partitions of projective spaces. The book describes general partitions of finite and infinite vector spaces, including Sperner spaces, focal-spreads, and their associated geometries. Since retraction groups provide quasi-subgeometry and subgeometry partitions of projective spaces, the author thoroughly discusses subgeometry partitions and their construction methods. He also features focal-spreads as partitions of vector spaces by subspaces. In addition to presenting many new examples of finite and infinite parallelisms, the book shows that doubly transitive or transitive t-parallelisms cannot exist unless the parallelism is a line parallelism. Along with the author’s other three books (Subplane Covered Nets, Foundations of Translation Planes, Handbook of Finite Translation Planes), this text forms a solid, comprehensive account of the complete theory of the geometries that are connected with translation planes in intricate ways. It explores how to construct interesting parallelisms and how general spreads of vector spaces are used to study and construct subgeometry partitions of projective spaces.