Bubble Lattices I: Structure

TL;DR

This paper introduces the bubble lattice, an extension of the shuffle lattice, providing a lattice-theoretic explanation for its properties and establishing its structural characteristics and connections to other known lattices.

Contribution

It defines and analyzes the bubble lattice, extending the shuffle lattice, and demonstrates its extremality, constructability, and relation to the Hochschild lattice.

Findings

The bubble lattice is extremal and constructable by interval doublings.

It generalizes the Hochschild lattice.

Provides a local and global characterization of the bubble lattice.

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.