CHAPTER 3

Foundations of Incidence Geometry

Francis BUEKENHOUT

CP 216, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium

Contents

1. Different viewpoints

2. The block space approach

2.1. Block spaces in general

2.2. Line spaces

2.3. Hyperplane spaces

2.4. Closure spaces

2.5. Other inspiration

3. The graph theoretic approach

3.1. Geometries

3.2. Special morphisms

3.3. Diagrams

3.4. Linear diagrams

3.5. Apartments

3.6. Shadows

3.7. The intersection property

3.8. Complexes

3.9. Chamber systems

4. The functional approach

4.1. Functional spaces

4.2. Join geometries and convexity spaces

4.3. André geometry

5. Incidence with other structure

5.1. Parallelism on block spaces

5.2. Hjelmslev--Klingenberg geometry

5.3. Incidence groups

5.4. Half ordered planes

References