Interval hypergraphic lattices

For a hypergraph H on [n], the hypergraphic poset P_H is the transitive closure of the oriented skeleton of the hypergraphic polytope △_H (the Minkowski sum of the standard simplices △_H for all H∈H). Hypergraphic posets include the weak order for the permutahedron (when H is the complete graph on [n]) and the Tamari lattice for the associahedron (when H is the set of all intervals of [n]), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of [n]. We characterize the interval hypergraphs I for which P_I is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order.