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The Lattice of Cyclic Flats of a Matroid

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arxiv math/0505689 v2 pith:NLVI64I7 submitted 2005-05-31 math.CO

classification math.CO
keywords cycliclatticeflatsmatroidmatroidsfunctioncircuitsclasses
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A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r on Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hepp's bound for Feynman graphs and matroids

    math-ph 2019-08 conditional novelty 8.0 of 10

    The Hepp bound, a rational matroid invariant from tropicalizing the Feynman period integral, provably respects all known graph period symmetries and correlates strongly with actual periods.

  2. Graph theoretic properties of Speyer's matroid polynomial $g_M(t)$

    math.CO 2025-06 accept novelty 7.0 of 10

    For graphic and cographic matroids, the derivative g'_M(-1) equals (-1)^{c(M)-1} c(M), and computational data suggests many new properties of the coefficient N2.

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