Isomorphisms between Lattices of Linear Subspaces Which are Induced by Isometries

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


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 k̄ and τ: 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.


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

  1. [1]
    R. Baer, Linear Algebra and Projective Geometry. Academic Press, New York 1952.Google Scholar
  2. [2]
    H. Gross, On Witt’s Theorem in the Denumerably Infinite Case. Math. Ann. 170 (1967) 145–165.CrossRefGoogle Scholar
  3. [3]
    H. Gross, Isomorphisms between lattices of linear subspaces which are induced by Isometries. J. Algebra 49 (1977) 537–546.CrossRefGoogle Scholar
  4. [4]
    H. Gross and H.A. Keller, On the definition of Hilbert Space. manuscripta math. 23 (1977) 67–90.CrossRefGoogle 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. Tůma, 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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