Adjoints of oriented matroids

Abstract

An adjoint of a geometric latticeG is a geometric lattice\(\tilde G\) of the same rank into which there is an embeddinge mapping the copoints ofG onto the points of\(\tilde G\). In this paper we introduce oriented adjoints and prove that they can be embedded into the extension lattice of oriented matroids.

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

References

  1. [1]

    L. J. Billera andB. S. Munson,Polarity and inner products in oriented matroids, Ithaca: Cornell University, School of Operations Research, (1981).

    Google Scholar 

  2. [2]

    R. E. Bixby andC. Coullard, Adjoints of binary matroids.European Journal of Combinatorics, to appear.

  3. [3]

    R. G. Bland, A combinatorial abstraction of linear programming.Journal of Combinatorial Theory B 23 (1977), 33–57.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    R. G. Bland andM. Las Vergnas, Orientability of matroids,Journal of Combinatorial Theory B 24 (1978), 94–123.

    MATH  Article  Google Scholar 

  5. [5]

    A. C. Chéung, Adjoints of a geometry,Canadian Math. Bul. 17 (1974), 363–365.

    MATH  Google Scholar 

  6. [6]

    R. Cordovil, Oriented matroids of rank three and arrangements of pseudolines, Ann. Discr. Math.17 (1933), 219–233.

    Google Scholar 

  7. [7]

    J. Folkman andL. Lawrence, Oriented matroids,Journal of Combinatorial Theory B 25 (1975), 199–236.

    Article  MathSciNet  Google Scholar 

  8. [8]

    M. Las Vergnas, Extensions ponctuelles d’une géométric combinatoire orientée,Acte du Coll Int CNRS,260, (1976), 263–268.

    Google Scholar 

  9. [9]

    M. Las Vergnas, Convexity in oriented matroids,Journal of Combinatorial Theory B 29 (1980), 231–243.

    MATH  Article  Google Scholar 

  10. [10]

    A. Mandel,Topology of oriented matroids. University of Waterloo, Department of Combinatorics and Optimization (Dissertation), 1981.

  11. [11]

    B. Spellman-Munson,Face lattices of oriented matroids. Ithaca: Cornell University (Dissertation), 1981.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Supported by Sonderforschungbereich 21 (DFG)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bachem, A., Kern, W. Adjoints of oriented matroids. Combinatorica 6, 299–308 (1986). https://doi.org/10.1007/BF02579255

Download citation

AMS subject classification (1980)

  • 05 B 35