The lattice of cyclic flats of a matroid

Anna de Mier

Abstract

The lattice of flats of a matroid is a well-understood object: it is a geometric lattice, and every geometric lattice is the lattice of flats of a matroid. In this talk we focus on a particular type of flats, cyclic flats, which also give rise to a lattice. A flat of a matroid is called cyclic if it is a union of circuits. It is easy to check that cyclic flats form a lattice under inclusion. But this lattice is far from having the "nice" properties that the lattice of flats has; for instance, it is not necessarily geometric and all maximal chains need not have the same length. We show that in fact every lattice is the lattice of cyclic flats of some matroid (and moreover, of a matroid that is both transversal and cotransversal). A matroid is uniquely determined by the set of its cyclic flats together with their ranks. We give a necessary and sufficient condition for a family of sets $\mathcal{Z}$ and a function $\rho: \