Lattices Related to Extensions of Presentations of Transversal Matroids

For a presentation A of a transversal matroid M , we study the ordered set TA of single-element transversal extensions of M that have presentations that extend A; extensions are ordered by the weak order. We show that TA is a distributive lattice, and that each finite distributive lattice is isomorphic to TA for some presentation A of some transversal matroid M . We show that TA ∩ TB, for any two presentations A and B of M , is a sublattice of both TA and TB. We prove sharp upper bounds on |TA| for presentations A of rank less than r(M) in the order on presentations; we also give a sharp upper bound on |TA ∩ TB|. The main tool we introduce to study TA is the lattice LA of closed sets of a certain closure operator on the lattice of subsets of {1, 2, . . . , r(M)}.