Ornamentation lattices and intreeval hypergraphic lattices
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911088552968192 |
|---|---|
| author | Abram, Antoine Bastidas, Jose Gélinas, Félix Pilaud, Vincent Sack, Andrew |
| author_facet | Abram, Antoine Bastidas, Jose Gélinas, Félix Pilaud, Vincent Sack, Andrew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01606 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ornamentation lattices and intreeval hypergraphic lattices Abram, Antoine Bastidas, Jose Gélinas, Félix Pilaud, Vincent Sack, Andrew Combinatorics 06B99, 06B10, 52B11, 52B12 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. |
| title | Ornamentation lattices and intreeval hypergraphic lattices |
| topic | Combinatorics 06B99, 06B10, 52B11, 52B12 |
| url | https://arxiv.org/abs/2508.01606 |