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aequationes mathematicae

, Volume 16, Issue 1–2, pp 37–50 | Cite as

Uniform binary geometries

  • Martin Aigner
Research papers

Abstract

In a geometric lattice every interval can be mapped isomorphically into an upper interval (containing 1) by a strong map. A natural question thus arises as to what extent certain assumptions on the “upper interval structure” determine the whole lattice. We consider conditions of the following sort: that above a certain levelm any two upper intervals of the same length be isomorphic. This property, called uniformity, is studied for binary geometries. The geometries satisfying the strongest uniformity condition (m = 1) are determined (except for one open case). As is to be expected the corresponding problem for lower intervals is easier and is solved completely.

AMS (1970) subject classification

Primary 05B25, 05B35 Secondary 06A25, 06A30 

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References

  1. [1]
    Aigner, M.,Reconstruction of Combinatorial Geometries. (1972), preprint.Google Scholar
  2. [2]
    Aigner, M.,Uniformität des Verbandes der Partitionen. Math. Ann.207 (1974), 1–22.Google Scholar
  3. [3]
    Baer, R.,Gruppentheoretische Begründung der Geometrie. Vorlesung Univ. Tübingen, 1953.Google Scholar
  4. [4]
    Birkhoff, G.,Lattice Theory. A.M.S. Colloq. Publ.25 (1940, 1948, 1967).Google Scholar
  5. [5]
    Crapo, H. H.,Single element extensions of Matroids. J. Res. Nat. Bur. Standards Sect. B.69 (1965), 55–65.Google Scholar
  6. [6]
    Crapo, H. H. andRota, G. C.,On the foundations of combinatiorial theory. Combinational Geometries, M.I.T. Press, 1969.Google Scholar
  7. [7]
    Dilworth, R. P. andGreene, C. A.,A counterexample to the generalization of Sperner's Theorem. J. Combinatorial Theory Ser. A.10 (1971), 18–21.Google Scholar
  8. [8]
    Dowling, T. A.,A q-analog of the partition lattice. Inst. Stat. Mimes Ser., University North Carolina, Nr. 779 (1971).Google Scholar
  9. [9]
    Edmonds, J., Murty, U. S. R. andYoung, P.,Matroid designs. Combinatorial Math. and Appl., 2nd Chapel Hill Conference, 1970, 498–542.Google Scholar
  10. [10]
    Higgs, D. H.,Geometry. Seminar Notes, University of Waterloo, 1966.Google Scholar
  11. [11]
    Murty, U. S. R.,Matroids with the Sylvester Property. Aequationes Math.4 (1970), 44–50.Google Scholar
  12. [12]
    Murty, U. S. R.,Equicardinal Matroids. J. Combinatorial Theory Ser. B.11 (1971), 120–126.Google Scholar
  13. [13]
    Ore, O.,Theory of equivalence relations. Duke Math. J.9 (1942), 573–627.Google Scholar
  14. [14]
    Wille, R.,Verbandstheoretische Charakterisiesung n-stufiger Geometrien. Arch. Math.18 (1967), 465–468.Google Scholar

Copyright information

© Birkhäuser Verlag 1977

Authors and Affiliations

  • Martin Aigner
    • 1
  1. 1.II. Mathematisches InstitutFreie Universität Berlin1 Berlin 33Germany

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