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Loops of Clutters

  • Michel Marie Deza
  • Komei Fukuda
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)

Abstract

For a clutter \( (C) \) on a finite set E, let \( f(C) \) be the clutter of maximal subsets of E not containing any member of \( (C) \). The loop \( L(C)\) of a clutter C is the finite sequence of clutters \( C,f(C),f^2 (C), \ldots ,f^t (C), \), where t (the length of the loop) is the minimum positive integer with \( f^t (C)\; = \;C. \). Our motivation of the study of the loop lies on the fact that when \( C \) is the set of circuits of a matroid, the loop contains other critical information associated with the matroid, e.g., the set of bases \( f (C) \) and the set of hyperplanes \( f^2 (C). \) We investigate various properties of the loops, in particular, the possible lengths for fixed |E|, the dualities and the symmetries. Our preliminary investigations indicate that there are many interesting problems on the loops whose resolution may provide us with a new insight into Sperner Theory, Matroid Theory and Extremal Set Theory.

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Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • Michel Marie Deza
    • 1
  • Komei Fukuda
    • 2
  1. 1.ParisFrance
  2. 2.Department of Information SciencesTokyo Institute of TechnologyOh-okayama, Meguro-ku, TokyoJapan

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