  An extension of the Tamari lattice to the multiplihedra is discussed, along
with projections to the composihedra and the Boolean lattice. The multiplihedra
and composihedra are sequences of polytopes that arose in algebraic topology
and category theory. Here we describe them in terms of the composition of
combinatorial species. We define lattice structures on their vertices, indexed
by painted trees, which are extensions of the Tamari lattice and projections of
the weak order on the permutations. The projections from the weak order to the
Tamari lattice and the Boolean lattice are shown to be different from the
classical ones. We generalize the Tamari lattice to graph tubings--as is also
described by Ronco. We review how lattice structures often interact with the
Hopf algebra structures, following Aguiar and Sottile who discovered the
applications of Mobius inversion on the Tamari lattice to the Loday-Ronco Hopf
algebra.
