Extending the Tamari Lattice to Some Compositions of Species

COMPOSITIONS OF SPECIES STEFAN FORCEY Abstract. 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 de ne 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 di erent from the classical ones. 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.