Projective and combinatorial geometries

Abstract

A combinatorial geometry is a pair ( X , ℱ) where X is a set of points and where ℱ is a family of subsets of X called flats such that ℱ is closed under (pairwise) intersection, there are no infinite chains in the poset ℱ, ℱ contains the empty set, all singletons { x }, x ∈ X , and the set X itself, for every flat E ∈ ℱ, E ≠ X , the flats that cover E in F partition the remaining points. Here, F covers E in F means that E, F ∈ ℱ, but that does not hold for any G ∈ ℱ. This latter property should be familiar to the reader from geometry: the lines that contain a given point partition the remaining points; the planes that contain a given line partition the remaining points. A trivial example of a geometry consists of a finite set X and all subsets of X as the flats. This is the Boolean algebra on X . We remark that (1) and (2) imply that ℱ is closed under arbitrary intersection. Example 23.1. Every linear space (as introduced in Chapter 19) gives us a combinatorial geometry on its point set X when we take as flats φ, all singletons {{ x } : x ∈ X }, all lines, and X itself. The fact that the lines on a given point partition the remaining points is another way of saying that two points determine a unique line.