Abstract
A geometry is of degree n (where n is any non-negative integer) if one and hence all of the 14 equivalent conditions of 4.4.1 and 4.4.4 are satisfied, and in particular if for any subspaces E, F the equation r(E ∨ F) + r(E ∧ F) = r(E) + r(F) holds provided that r(E ∧ F) ≥ n. The geometries of degree 0 are exactly the projective geometries. Among the geometries of degree 1 one finds the affine geometries and also the projective geometries (we remark that a geometry of degree n is trivially of degree n + 1). Within the geometries of degree 1 the projective geometries can be characterized by an axiom requiring that parallel lines must be equal, the affine geometries by an axiom requiring that for any line δ and any point p there exists a unique line parallel to δ containing p.
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© 2000 Springer Science+Business Media Dordrecht
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Faure, CA., Frölicher, A. (2000). Geometries of Degree n . In: Modern Projective Geometry. Mathematics and Its Applications, vol 521. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9590-2_5
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DOI: https://doi.org/10.1007/978-94-015-9590-2_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5544-6
Online ISBN: 978-94-015-9590-2
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