On reid’s 3-simplicial matroid theorem


In this paper we prove the following result of Ralph Reid (which was never published nor completely proved).

Theorem. Let M be a matroid coordinatizable (representable) over a prime field F. Then there is a 3-simplicial matroid M′ over F which is a series extension of M.

The proof we give is different from the original proof of Reid which uses techniques of algebraic topology. Our proof is constructive and uses elementary matrix operations.

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  1. [1]

    R. Cordovil, Sur les géométries simpliciales,C. R. Acad. Sci. Paris Sér. A,286 (1978), 1219–1222.

    MATH  MathSciNet  Google Scholar 

  2. [2]

    R. Cordovil, Representation over a Field of Full Simplicial Matroids,Europ. J. Combinatorics 1 (1980), 201–205.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    H. H. Crapo andG.-C. Rota,On the Foundations of Combinatorial Theory: Combinatorial Geometries, M.I.T. Press, Cambridge, Mass., 1970.

    Google Scholar 

  4. [4]

    H. H. Crapo andG.-C. Rota, Simplicial geometries,Proc. Symp. Pure Math.,XIX.,Combinatorics, American Mathematical Society Publication, 1971.

  5. [5]

    R. Reid, Polyhedral and Simplicial Geometries,unpublished.

  6. [6]

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

    Google Scholar 

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Cordovil, R. On reid’s 3-simplicial matroid theorem. Combinatorica 2, 135–141 (1982). https://doi.org/10.1007/BF02579311

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AMS Subject Classification (1980)

  • 05 B 35