Subplane Covered Nets

This work confronts the question of geometric processes of derivation, specifically the derivation of affine planes - keying in on construction techniques and types of transformations in which lines of a newly-created plane can be understood as subplanes of the original plane. The book provides a theory of subplane covered nets without restriction to the finite case or imposing commutativity conditions.

Preface

iii

A Brief Overview

1

(6)

Projective Geometries

7

(10)

Projective and Affine Geometries

9

(5)

Projective and Affine Planes

14

(3)

Beginning Derivation

17

(10)

Affine Forms

19

(2)

Affine Space as a Vector Space

21

(1)

PG(3,q) and the Regulus

21

(2)

The Standard Vector Space Form of a Regulus

22

(1)

The Regulus Question

23

(1)

Infinite Derivation

23

(4)

Spreads

27

(14)

Affine Planes with a Transitive Translation Group

27

(4)

The Fundamental Theorem of Translation Planes

31

(2)

The Basis Decomposition Theorem

33

(8)

Some Examples

37

(2)

Partial Spreads

39

(1)

Derivable Partial Spread

40

(1)

Derivable Nets

41

(16)

Coordinatization

42

(5)

Coordinates of a Net

43

(1)

The Ternary Function

43

(1)

Addition

44

(1)

Multiplication

44

(1)

Extension to an Affine Plane and Linearity

45

(1)

The Question of Linearity

45

(2)

Coordinates for Translation Planes and Their Duals

47

(10)

Kernel Mappings and Kernel of the Quasifield

50

(1)

Dual Affine Planes

51

(1)

Dual Translation Planes and Derivation

52

(1)

Chains and Semi-Translation Planes

53

(4)

The Hughes Planes

57

(8)

The Affine Hughes Planes

60

(5)

Desarguesian Planes

65

(14)

Moufang Planes

67

(12)

Coordinate Structures for Moufang Planes

69

(10)

Pappian Planes

79

(6)

Reguli in Pappian Spreads

80

(5)

Characterizations Of Geometries

85

(12)

The Fundamental Theorem of Projective Geometry

85

(3)

Projective and Affine Geometries

88

(9)

Derivable Nets And Geometries

97

(16)

The Characterization Theorem

102

(11)

Cofman's Theorem (for Derivable Nets)

104

(3)

A Quilt of Affine Geometries

107

(6)

Structure Theory For Derivable Nets

113

(18)

The Collineation Group of a Derivable Net

114

(12)

The Translation Group of a Derivable Net

116

(2)

A Representation for the Translation Group

118

(4)

The Group of Quasi-Skew and Skew Perspectivities

122

(4)

The Structure Theorem for Derivable Nets

126

(1)

The Answer to the Regulus Question

127

(1)

The Solution to the Coordinates Problem

127

(4)

Dual Spreads And Baer Subplanes

131

(12)

Coordinate-Free Derivation and the Extension Question

137

(6)

Coordinate-Free Derivation

141

(2)

Derivation As A Geometric Process

143

(6)

The Annihilator Mapping

143

(2)

A `Geometric' Interpretation

145

(4)

Embedding

149

(16)

Embedding Nets into Affine Space

151

(3)

Embedding Nets into Projective Space

154

(2)

Co-dimension-Two and Pseudo-Regulus Nets

156

(9)

Embedded Nets Are Pseudo-Regulus Nets

163

(2)

Classification Of Subplane Covered Nets

165

(36)

Sketch of the Embedding

167

(1)

The Share Two Theorem

168

(9)

The Derivable Net Substructures

177

(9)

The Subnet Theorem

183

(3)

The Parallel Classes Are Affine Spaces

186

(5)

The Projective Space

191

(10)

The Classification Theorem

195

(3)

The Theorem on Pseudo-Regulus Nets

198

(3)

Subplane Covered Affine Planes

201

(14)

Regular Spreads

201

(5)

The Affine Little Reidemeister Condition

206

(4)

Embedded Nets

210

(3)

Subplane Covered Translation Planes

213

(2)

Direct Products

215

(18)

The Reconstruction Theorem

217

(11)

Desarguesian Products

225

(3)

Left Spreads Produce Right Partial Spreads

228

(1)

Left Spreads Produce Left Partial Spreads

229

(4)

2-fold Products

231

(2)

Parallelisms

233

(12)

Desarguesian Parallelisms

233

(2)

When the Vector Space Is 4-dimensional

235

(2)

Desarguesian Left Parallelisms Construct Right Spreads

237

(2)

An Infinite Class of `Derived' Parallelisms

239

(4)

The Parallelisms in PG(3,2)

243

(2)

Partial Parallelisms With Deficiency

245

(2)

Deficiency One

247

(2)

Partial Dual Parallelisms

249

(2)

Baer Extensions

251

(24)

Point-Baer Collineations

251

(5)

Notes on the Non-Commutative Case

256

(6)

Left and Right in Direct Product Nets

259

(2)

When K Is Ko

261

(1)

Point-Baer Subplanes in Vector Space Nets

262

(13)

Three Point-Baer Subplanes

263

(5)

When There Are Two Point-Baer Subplanes

268

(2)

The Space Is Decomposable by a Point-Baer Subplane

270

(5)

Translation Planes Admitting Baer Groups

275

(18)

General Spreads with Baer Groups

275

(10)

Translation Planes with Spreads in PG(3,K)

285

(8)

Spreads Covered By Pseudo-Reguli

293

(16)

Pseudo-Reguli

293

(7)

Conical and Ruled Spreads over Skewfields

300

(9)

Conical And Ruled Planes Over Fields

309

(12)

Conical Flocks

309

(4)

Hyperbolic Flocks

313

(8)

Spreads Which Are Dual Spreads

321

(20)

Chains and Semi-translation Planes Revisited

326

(3)

Partial Flocks Of Deficiency One

329

(1)

Point-Baer Subplanes in Spreads of PG(3, K)

329

(2)

Deficiency-One Partial Conical Flocks

331

(5)

Deficiency-One Partial Hyperbolic Flocks

336

(5)

Skew-Hall Planes

341

(12)

Double-covers

347

(3)

Postscript

350

(3)

Bibliography

353

(6)

Index

359

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Subplane Covered Nets

Summary

Table of Contents

Supplemental Materials