Hochschild lattices and shuffle lattices

In his study of a Hochschild complex arising in connection with the free loop fibration, S. Saneblidze defined freehedron, certain polytope constructed via truncation process from hypercube. It was recently conjectured by F. Chapoton and proven C. Combe that orientation 1-skeleton freehedron carries lattice structure. The resulting dubbed it is interval constructable extremal. These properties allow for definition three associated structures: Galois graph, canonical join core label order. this article, we characterize these structures. We exhibit an isomorphism order to particular shuffle Greene. also uncover enumerative between lattice, extension its poset irreducibles freehedron. connections nicely parallel situation surrounding better-known Tamari lattices, noncrossing partition lattices associahedra.