The Lattice of Cyclic Flats of a Matroid |
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Authors: | Joseph E Bonin Anna de Mier |
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Institution: | (1) Department of Mathematics, The George Washington University, Washington, D.C. 20052, USA;(2) Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, United Kingdom |
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Abstract: | 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 of sets and a function to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to . 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.
Received May 29, 2005 |
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Keywords: | :" target="_blank">: matroid cyclic flat Dilworth’ s embedding theorem nested matroid free product Tutte polynomial |
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