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Bubble Lattices I: Structure

Thomas McConville, Henri Mühle

Abstract

C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. We characterize the bubble lattice both locally (via certain transformations of shuffle words) and globally (using a notion of inversion set). We then prove that the bubble lattice is extremal and constructable by interval doublings. Lastly, we prove that our bubble lattice is a generalization of the Hochschild lattice studied earlier by Chapoton, Combe and the second author.

Bubble Lattices I: Structure

Abstract

C. Greene introduced the shuffle lattice as an idealized model for DNA mutation and discovered remarkable combinatorial and enumerative properties of this structure. We attempt an explanation of these properties from a lattice-theoretic point of view. To that end, we introduce and study an order extension of the shuffle lattice, the bubble lattice. We characterize the bubble lattice both locally (via certain transformations of shuffle words) and globally (using a notion of inversion set). We then prove that the bubble lattice is extremal and constructable by interval doublings. Lastly, we prove that our bubble lattice is a generalization of the Hochschild lattice studied earlier by Chapoton, Combe and the second author.
Paper Structure (15 sections, 37 theorems, 28 equations, 8 figures, 1 table)

This paper contains 15 sections, 37 theorems, 28 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Posets
  4. Lattices
  5. Semidistributive Lattices
  6. Lattices Constructable by Interval Doublings
  7. Trim Lattices
  8. Shuffle Words
  9. The Order and Cover Relation of $\textbf{Bub}(m,n)$
  10. Lattice Properties of $\textbf{Bub}(m,n)$
  11. Joins in $\textbf{Bub}(m,n)$
  12. Extremality of $\textbf{Bub}(m,n)$
  13. Interval-Constructability of $\textbf{Bub}(m,n)$
  14. Relation to Hochschild Lattices
  15. The Galois Graph of $\textbf{Bub}(m,n)$

Key Result

Theorem 1.1

For $m,n\geq 0$, the bubble lattice $\textbf{Bub}(m,n)$ is a lattice. Moreover, $\textbf{Bub}(m,n)$ is extremal, semidistributive and constructable by interval doublings.

Figures (8)

  • Figure 1: Two posets of shuffle words.
  • Figure 2: The covering pairs $(a_{1},b_{1})$ and $(b_{2},c_{2})$ are perspective. The covering pair $(a_{2},b_{2})$ is not perspective to any other covering pair.
  • Figure 3: Doubling the poset on the left by the highlighted interval yields the poset on the right.
  • Figure 4: The poset $\textbf{Bub}(2,1)$ labeled by $\lambda$.
  • Figure 5: The poset $\mathbf{S}_{4,3}$.
  • ...and 3 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 2.1: freese:free*Theorem 2.24
  • Proposition 2.2: barnard:canonical*Lemma 19
  • Lemma 2.3: muehle:distributive*Lemma 3.3
  • Lemma 2.4: freese:free*Corollary 2.55
  • Theorem 2.5: garver.mcconville:oriented_tree*Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8: day:characterizations*Lemma 4.2
  • Lemma 2.9: garver.mcconville:oriented_tree*Lemma 2.6
  • ...and 57 more