Ornamentation lattices and intreeval hypergraphic lattices

Given a directed graph D with transitive closure tc(D) and path hypergraph P(D), we study the connections between the (acyclic) reorientation poset of tc(D), the (acyclic) sourcing poset of P(D), and the (acyclic) ornamentation poset of D. Geometrically, the acyclic reorientation poset of tc(D) (resp. the acyclic sourcing poset of P(D)) is the transitive closure of the skeleton of the graphical zonotope of tc(D) (resp. of the hypergraphic polytope of 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 P(D) is isomorphic to the ornamentation lattice of D, and that they form a lattice quotient of the acyclic reorientation lattice of 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 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 P(D) whose acyclic sourcing poset is a lattice.