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