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Lattices related to extensions of presentations of transversal matroids

Joseph E. Bonin

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

Lattices related to extensions of presentations of transversal matroids

Abstract

For a presentation of a transversal matroid , we study the set of single-element transversal extensions of that have presentations that extend ; we order these extensions by the weak order. We show that is a distributive lattice, and that each finite distributive lattice is isomorphic to for some presentation of some transversal matroid . We show that , for any two presentations and of , is a sublattice of both and . We prove sharp upper bounds on for presentations of rank less than in the order on presentations; we also give a sharp upper bound on . The main tool we introduce to study is the lattice of closed sets of a certain closure operator on the lattice of subsets of .

Paper Structure

This paper contains 10 sections, 35 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. A closure operator and two isomorphic distributive lattices
  4. The lattice $L_\mathcal{A}$
  5. The lattice $T_\mathcal{A}$
  6. The sets in $L_\mathcal{A}$
  7. The intersection of $T_\mathcal{A}$ and $T_\mathcal{B}$
  8. How to get any finite distributive lattice
  9. Irreducible elements
  10. Applications

Key Result

Lemma 2.1

Let $M$ be $M[\mathcal{A}]$ with $\mathcal{A} =(A_i\,:\,i\in [r])$. For any subset $X$ of $E(M)$, the restriction $M|X$ is transversal and $(A_i\cap X\,:\,i\in [r])$ is a presentation of $M|X$.

Figures (5)

  • Figure 1: Two presentations $\mathcal{A}$ of a transversal matroid $M$, along with the associated lattices $L_\mathcal{A}$.
  • Figure 2: A transversal matroid whose minimal presentations are also maximal. The points $r,s,t,u$ are freely in the shaded plane.
  • Figure 3: The presentations and the meet of the extensions discussed in Example 2. In the first figure, $g$ is in no proper face of the simplex; in the second, $h$ is in no proper face.
  • Figure 4: An example, for $U_{6,7}$, of the construction of $\mathcal{B}$ in the proof of Theorem \ref{['thm:alldistributivelattices']}, with $L$ on the left and the sets $I_0$ on the right. The presentation has $B_1=\{1,2,3,4,5,6,7\}$, $B_2=B_3=\{2,3,6,7\},$$B_4=B_5=\{4,5,6,7\}$, and $B_6=\{6,7\}$.
  • Figure 5: The induced order on the irreducibles of $L_{i-1}$.

Theorems & Definitions (52)

  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Corollary 2.9
  • Lemma 2.10
  • ...and 42 more