Loops of Clutters
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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