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Combinatorica

, Volume 5, Issue 3, pp 237–239 | Cite as

A desarguesian theorem for algebraic combinatorial geometries

  • B. Lindström
Article

Abstract

The points of an algebraic combinatorial geometry are equivalence classes of transcendentals over a fieldk; two transcendentals represent the same point when they are algebraically dependent overk. The points of an algebraically closed field of transcendence degree two (three) overk are the lines (resp. planes) of the geometry.

We give a necessary and sufficient condition for two coplanar lines to meet in a point (Theorem 1) and prove the converse of Desargues’ theorem for these geometries (Theorem 2). A corollary: the “non-Desargues” matroid is non-algebraic.

The proofs depend on five properties (or postulates). The fifth of these is a deep property first proved by Ingleton and Main [3] in their paper showing that the Vámos matroid is non-algebraic.

AMS subject classification (1980)

05 B 35 50 D 50 12 F 20 

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References

  1. [1]
    J. R. Bastida,Field Extensions and Galois Theory, Addison—Wesley 1984.Google Scholar
  2. [2]
    H. Crapo andG.-C. Rota,Combinatorial Geometries (prel. ed.) M. I. T. Press, 1970.Google Scholar
  3. [3]
    A. W. Ingleton andR. A. Main, Non-algebraic matroids exist,Bull. London Math. Soc. 7 (1975), 144–146.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    B. Lindström, The non-Pappus Matroid is algebraic,Ars Combinatoria 16 B (1983), 95–96.Google Scholar
  5. [5]
    B. Lindström, A simple non-algebraic matroid of rank three,Utilitas Mathematica 25 (1984), 95–97.zbMATHMathSciNetGoogle Scholar
  6. [6]
    L. Lovász,private communication.Google Scholar
  7. [7]
    S. Mac Lane, A lattice formulation for transcendence degrees andp-bases,Duke Math. J. 4 (1938), 455–468.CrossRefMathSciNetGoogle Scholar
  8. [8]
    D. J. A. Welsh,Matroid Theory, Academic Press, 1976.Google Scholar

Copyright information

© Akadémiai Kiadó 1985

Authors and Affiliations

  • B. Lindström
    • 1
  1. 1.Department of MathematicsUniversity of StockholmStockholmSweden

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