Interval hypergraphic lattices

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_{\mathbb{H}}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope ${\mathrm{△}}_{\mathbb{H}}$ (the Minkowski sum of the standard simplices ${\mathrm{△}}_H$ for all $H\in \mathbb{H}$). Hypergraphic posets include the weak order for the permutahedron (when $\mathbb{H}$ is the complete graph on $[n]$) and the Tamari lattice for the associahedron (when $\mathbb{H}$ 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 $\mathbb{I}$ for which $P_{\mathbb{I}}$ is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order.