Abstract
A at of a matroid is cyclic if it is a union of circuits. The cyclic ats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic ats. In particular, we show that every lattice is isomorphic to the lattice of cyclic ats of a matroid. We give a necessary and sufcient condition for a latticeZ of sets and a function r : Z ! Z to be the lattice of cyclic ats of a matroid and the restriction of the corresponding rank function to 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 dene 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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