Ornamentation lattices and intreeval hypergraphic lattices

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