A desarguesian theorem for algebraic combinatorial geometries

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    J. R. Bastida,Field Extensions and Galois Theory, Addison—Wesley 1984.

  2. [2]

    H. Crapo andG.-C. Rota,Combinatorial Geometries (prel. ed.) M. I. T. Press, 1970.

  3. [3]

    A. W. Ingleton andR. A. Main, Non-algebraic matroids exist,Bull. London Math. Soc. 7 (1975), 144–146.

    MATH  Article  MathSciNet  Google 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.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    L. Lovász,private communication.

  7. [7]

    S. Mac Lane, A lattice formulation for transcendence degrees andp-bases,Duke Math. J. 4 (1938), 455–468.

    Article  MathSciNet  Google Scholar 

  8. [8]

    D. J. A. Welsh,Matroid Theory, Academic Press, 1976.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lindström, B. A desarguesian theorem for algebraic combinatorial geometries. Combinatorica 5, 237–239 (1985). https://doi.org/10.1007/BF02579367

Download citation

AMS subject classification (1980)

  • 05 B 35
  • 50 D 50
  • 12 F 20