The Lattice of Cyclic Flats of a Matroid

Details The Lattice of Cyclic Flats of a Matroid The Lattice of Cyclic Flats of a Matroid

2005-05-31

arxiv.org/abs/math/0505689v2

Mathematics

Combinatorics

15 pages, 1 figure. The new version addresses earlier work by Julie Sims that the authors learned of after submitting the firs

Scientific paper

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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