Interval hypergraphic lattices
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909944472666112 |
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| author | Bergeron, Nantel Pilaud, Vincent |
| author_facet | Bergeron, Nantel Pilaud, Vincent |
| contents | For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope $\triangle_\mathbb{H}$ (the Minkowski sum of the standard simplices $\triangle_H$ for all $H \in \mathbb{H}$). Hypergraphic posets include the weak order for the permutahedron (when $\mathbb{H}$ is the complete graph on $[n]$) and the Tamari lattice for the associahedron (when $\mathbb{H}$ is the set of all intervals of $[n]$), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of $[n]$. We characterize the interval hypergraphs $\mathbb{I}$ for which $P_\mathbb{I}$ is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interval hypergraphic lattices Bergeron, Nantel Pilaud, Vincent Combinatorics 06B99, 06D99, 52B11, 52B12 For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope $\triangle_\mathbb{H}$ (the Minkowski sum of the standard simplices $\triangle_H$ for all $H \in \mathbb{H}$). Hypergraphic posets include the weak order for the permutahedron (when $\mathbb{H}$ is the complete graph on $[n]$) and the Tamari lattice for the associahedron (when $\mathbb{H}$ is the set of all intervals of $[n]$), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of $[n]$. We characterize the interval hypergraphs $\mathbb{I}$ for which $P_\mathbb{I}$ is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order. |
| title | Interval hypergraphic lattices |
| topic | Combinatorics 06B99, 06D99, 52B11, 52B12 |
| url | https://arxiv.org/abs/2411.09832 |