Ornamentation lattices and intreeval hypergraphic lattices

TL;DR

This paper explores the relationships between various posets associated with directed graphs and hypergraphs, establishing lattice structures and polytopal realizations, especially for increasing trees, and addressing open questions in the field.

Contribution

It introduces new connections between reorientation, sourcing, and ornamentation posets, providing lattice and polytopal structures for these objects, particularly in the context of increasing trees.

Findings

The acyclic sourcing poset of the path hypergraph is isomorphic to the ornamentation lattice for increasing trees.

The ornamentation lattice forms a lattice quotient of the acyclic reorientation lattice.

Polytopal realizations of ornamentation lattices are constructed, answering open questions.

Abstract

Given a directed graph $D$ with transitive closure $\operatorname{tc}(D)$ and path hypergraph $\mathbb{P}(D)$, we study the connections between the (acyclic) reorientation poset of $\operatorname{tc}(D)$, the (acyclic) sourcing poset of $\mathbb{P}(D)$, and the (acyclic) ornamentation poset of $D$. Geometrically, the acyclic reorientation poset of $\operatorname{tc}(D)$ (resp. the acyclic sourcing poset of $\mathbb{P}(D)$) is the transitive closure of the skeleton of the graphical zonotope of $\operatorname{tc}(D)$ (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 $\operatorname{tc}(D)$. 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.