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The lattice of cyclic flats of a matroid. (English) Zbl 1145.05015

Summary: 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 \({\mathcal{Z}}\) of sets and a function \(r : {\mathcal{Z}} \rightarrow {\mathbb{Z}}\) to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to \({\mathcal{Z}}\). We apply this perspective to give an alternative view of the free product of matroids and we show how to compute the Tutte polynomial of the free product in terms of the Tutte polynomials of the constituent matroids. 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.

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices
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