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Combinatorial representation and convex dimension of convex geometries

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Abstract

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.

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References

  1. G. Birkhoff (1967) Lattice Theory, 3rd edn., Amer. Math. Soc. Coll. Publ. Vol. 25, Providence, RI.

  2. R. P. Dilworth (1950) A decomposition theorem for partially ordered sets, J. Ann. Math. 51, 161–166.

    Google Scholar 

  3. P. H. Edelman (1986) Abstract convexity and meet-distributive lattices, Combinatorics and Ordered Sets (I. Rival, Ed.), AMS Contemporary Mathematics, Vol. 57, 127–150.

  4. P. H. Edelman (1980) Meet-distributive lattices and the anti-exchange closure, Alg. Univ. 10, 290–299.

    Google Scholar 

  5. P. H. Edelman and R. E. Jamison (1985) The theory of convex geometries, Geom. Dedicata 19, 247–270.

    Google Scholar 

  6. B. Korte and L. Lovász (1985) Homomorphisms and Ramsey properties of antimatroids, Report No. 85364-OR, Institute für Ökonometric and Operations Research, Bonn.

    Google Scholar 

  7. D. Kelly and W. T. Trotter (1982) Dimension Theory for Ordered Sets, in Ordered Sets (I. Rival, ed.) D. Reidel, Dordrecht, pp. 171–211.

    Google Scholar 

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Author notes

  1. Paul H. Edelman

    Present address: Department of Mathematics, University of Minnesota, 55455, Minneapolis, MN, USA

Authors and Affiliations

  1. Department of Mathematics, Carnegie-Mellon University, 15213, Pittsburgh, PA, USA

    Paul H. Edelman

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Michael E. Saks

  3. Bell Communications Research, 07960, Morristown, NJ, USA

    Michael E. Saks

Authors
  1. Paul H. Edelman
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  2. Michael E. Saks
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Additional information

Communicated by D. Kelly

Supported in part by NSF grant no. DMS-8501948.

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Edelman, P.H., Saks, M.E. Combinatorial representation and convex dimension of convex geometries. Order 5, 23–32 (1988). https://doi.org/10.1007/BF00143895

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  • DOI: https://doi.org/10.1007/BF00143895

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