  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.
