Ornamentation lattices and intreeval hypergraphic lattices

Given a directed graph $D$ with transitive closure $\mathrm{tc}(D)$ and path hypergraph $\mathbb{P}(D)$, we study the connections between the (acyclic) reorientation poset of $\mathrm{tc}(D)$, the (acyclic) sourcing poset of $\mathbb{P}(D)$, and the (acyclic) ornamentation poset of $D$. Geometrically, the acyclic reorientation poset of $\mathrm{tc}(D)$ (resp. the acyclic sourcing poset of $\mathbb{P}(D)$) is the transitive closure of the skeleton of the graphical zonotope of $\mathrm{tc}(D)$ (resp. of the hypergraphic polytope of $\mathbb{P}(D)$) oriented in a linear direction. When $D$ is a rooted (or even unstarred) increasing tree, we show that the acyclic sourcing poset of $\mathbb{P}(D)$ is isomorphic to the ornamentation lattice of $D$, and that they form a lattice quotient of the acyclic reorientation lattice of $\mathrm{tc}(D)$. As a consequence, we obtain polytopal realizations of the ornamentation lattices of rooted (or even unstarred) increasing trees, answering an open question of C. Defant and A. Sack. When $D$ is an increasing tree, we show that the ornamentation lattice of $D$ is the MacNeille completion of the acyclic sourcing poset of $\mathbb{P}(D)$. Finally, still when $D$ is an increasing tree, we use the ornamentation lattice of $D$ to characterize the subhypergraphs of the path hypergraph $\mathbb{P}(D)$ whose acyclic sourcing poset is a lattice.