Modern Projective Geometry pp 25-53 | Cite as

# Projective Geometries and Projective Lattices

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## Abstract

There exist many equivalent axiomatic characterizations of projective geometries. The first two sections of this chapter present two such possibilities. The first one uses the ternary relation *collinear* on the set of points. The second one works with the operator that associates to each couple of points a, *b* the line through a and *a* ≠ *b* and the singleton {*a*} if a = *b*. Another approach, by means of the closure operator *C* that associates to an arbitrary set *A* of points the subspace *C*(*A*) generated by *A*, will be studied in Chapter 3. The first section also includes a series of examples. The most important one is the projective geometry associated to an arbitrary vector space *V*. This geometry will be denoted by *P*(*V*). The points of *P*(*V*) are the one-dimensional vector subspaces of *V*, three of them being called collinear if they lie in a vector subspace of dimension ≤ 2.

## Keywords

Irreducible Component Intersection Property Closure Operator Vector Space Versus Projective Geometry## Preview

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