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Diagram Geometry pp 201–251Cite as

Linear Geometries

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Abstract

In Example 1.4.9 we introduced the projective geometry PG(V) and in Example 1.4.10 the affine geometry AG(V) associated with a vector space V of finite dimension n. In Proposition 2.4.7 the geometry PG(V) was shown to have a linear Coxeter diagram A n−1, and in Proposition 2.4.10 the geometry AG(V) was shown to belong to the linear diagram Af n . We now turn our attention to the more general class of all geometries with a linear diagram. The shadow spaces on 1 of our motivating examples PG(V) and AG(V) are linear line spaces (in the sense that any two points are on a unique line; cf. Definition 2.5.13), and we will restrict ourselves mostly to geometries with this property. Within this class there are combinatorial structures such as matroids and Steiner systems.

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Authors and Affiliations

  1. Dépt. Mathématique, Université Libre de Bruxelles, Académie royale de Belgique, Bruxelles, Belgium

    Francis Buekenhout

  2. Dept. Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands

    Arjeh M. Cohen

Authors
  1. Francis Buekenhout
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  2. Arjeh M. Cohen
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Buekenhout, F., Cohen, A.M. (2013). Linear Geometries. In: Diagram Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34453-4_5

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