, Volume 5, Issue 1, pp 23–32 | Cite as

Combinatorial representation and convex dimension of convex geometries

  • Paul H. Edelman
  • Michael E. Saks


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.

AMS subject classifications (1980)

06A10 06B05 

Key words

Meet distributive lattices convex geometries convex dimension 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Birkhoff (1967) Lattice Theory, 3rd edn., Amer. Math. Soc. Coll. Publ. Vol. 25, Providence, RI.Google Scholar
  2. 2.
    R. P. Dilworth (1950) A decomposition theorem for partially ordered sets, J. Ann. Math. 51, 161–166.Google Scholar
  3. 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.Google Scholar
  4. 4.
    P. H. Edelman (1980) Meet-distributive lattices and the anti-exchange closure, Alg. Univ. 10, 290–299.Google Scholar
  5. 5.
    P. H. Edelman and R. E. Jamison (1985) The theory of convex geometries, Geom. Dedicata 19, 247–270.Google Scholar
  6. 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. 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

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • Paul H. Edelman
    • 1
  • Michael E. Saks
    • 2
    • 3
  1. 1.Department of MathematicsCarnegie-Mellon UniversityPittsburghUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA
  3. 3.Bell Communications ResearchMorristownUSA

Personalised recommendations