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Isomorphisms between Lattices of Linear Subspaces Which are Induced by Isometries

  • Herbert Gross
Part of the Progress in Mathematics book series (PM, volume 1)

Abstract

Let E be a vector space over the division ring k and L(E) the lattice of all linear subspaces of E. If Ē is a vector space over the division ring k and T: L(E); → L(Ē) a lattice isomorphism then by the Fundamental Theorem of Projective Geometry ([1] p. 44) τ is induced by a semilinear map T: E → Ē if we assume that dim E ≥ 3.

Keywords

Linear Subspace Lattice Versus Projective Geometry Division Ring Modular Lattice 
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 to Chapter IV

  1. [1]
    R. Baer, Linear Algebra and Projective Geometry. Academic Press, New York 1952.MATHGoogle Scholar
  2. [2]
    H. Gross, On Witt’s Theorem in the Denumerably Infinite Case. Math. Ann. 170 (1967) 145–165.MathSciNetCrossRefGoogle Scholar
  3. [3]
    H. Gross, Isomorphisms between lattices of linear subspaces which are induced by Isometries. J. Algebra 49 (1977) 537–546.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    H. Gross and H.A. Keller, On the definition of Hilbert Space. manuscripta math. 23 (1977) 67–90.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    C. Herrmann, On a condition sufficient for the distributivity of lattices of linear subspaces. To appear.Google Scholar
  6. [6]
    P. Pudlak and J. Túma, Yeast graphs and fermentation of algebraic lattices. Coll. Math. Soc. J. Bolyai, 14 (1976) Lattice Theory 301–342 ed. by A.P. Huhn and E.T. Schmidt, North Holland Publ. Company, Amsterdam.Google Scholar

Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Herbert Gross
    • 1
  1. 1.Mathematisches InstitutUniversität ZürichZürichSwitzerland

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