Extending the Tamari lattice to some compositions of species

TL;DR

This paper extends the Tamari lattice to multiplihedra and composihedra using combinatorial species, defining new lattice structures and projections that relate to algebraic topology and category theory.

Contribution

It introduces a novel extension of the Tamari lattice to multiplihedra and composihedra via combinatorial species, and explores their lattice and projection structures.

Findings

Defined lattice structures on vertices indexed by painted trees

Established new projections from the weak order to Tamari and Boolean lattices

Connected lattice structures with Hopf algebra applications

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 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.