Lattices related to extensions of presentations of transversal matroids

TL;DR

This paper investigates the structure of extensions of transversal matroids through lattice theory, showing that the set of single-element extensions forms a distributive lattice and characterizing their intersections and bounds.

Contribution

It establishes that the set of single-element transversal extensions forms a distributive lattice and characterizes their intersections and bounds, linking lattice theory with matroid extensions.

Findings

The set of extensions forms a distributive lattice.

Each finite distributive lattice corresponds to some presentation of a transversal matroid.

Sharp bounds are provided for the size of extension sets and their intersections.

Abstract

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 $\{1,2,\ldots,r(M)\}$.