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Geometries uniquely embeddable in projective spaces

Part II: Contributed Papers

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Part of the Lecture Notes in Mathematics book series (LNM,volume 893)

Abstract

The search for geometric lattices, which admit an embedding in a projective space satisfying universal or uniqueness properties, has been initiated by W.M. Kantor [2,3] (see also the survey [4]). A recent result of the author [8] leads to a unification and generalization of Kantor's theorems (see Theorem 1). A typical application of this generalization is the following (see Section III.D).

Let G be a geometric lattice which is not the union of two hyperplanes. Assume that each interval [p,1] of G, where p is a point, is isometrically embeddable in PG(m-1,K) for some field K, and that all embeddings of [L,1] in PG(m-2,K), where L is a line, are projectively equivalent, then G is isometrically embeddable in PG(m,K) (and its embeddings satisfy some uniqueness properties).

Keywords

  • Projective Space
  • Full Subcategory
  • Projective Geometry
  • Finite Rank
  • Embed Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Aigner M., Combinatorial theory, Springer-Verlag, New York 1979.

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  2. Kantor W.M., Dimension and embedding theorems for geometric lattices, J. Combin. Theory Ser. A 17 (1974), 173–195.

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  3. Kantor W.M., Envelopes of geometric lattices, J. Combin. Theory Ser. A 18 (1975), 12–26.

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  4. Kantor W.M., Some highly geometric lattices, Atti Conv. Lincei 17 (1976), 183–191.

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  5. Percsy N., Peongement de géométries, Thèse de doctorat, ULB, Bruxelles 1980.

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  6. Percsy N., Embedding geometric lattices in a projective space, Proc. 2nd Conb. on Finite Geometries and Designs, Cambridge U. Press, London 1981.

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  7. Percsy N., Locally embeddable geometries, to appear in Arch. Math. (1981).

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  8. Percsy N., Une condition nécessaire et suffisante de plongeabilité pour les treillis semi-modulaires, to appear in European J. Combin. (1981).

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  9. Percsy N., Sufficient embeddability conditions for geometric lattices, in preparation.

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  10. Welsh D.J.A., Matroid theory, Academic Press, London 1976.

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© 1981 Springer-Verlag

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Percsy, N. (1981). Geometries uniquely embeddable in projective spaces. In: Aigner, M., Jungnickel, D. (eds) Geometries and Groups. Lecture Notes in Mathematics, vol 893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091024

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  • DOI: https://doi.org/10.1007/BFb0091024

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11166-5

  • Online ISBN: 978-3-540-38639-1

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