Interval hypergraphic lattices

TL;DR

This paper investigates the lattice properties of hypergraphic posets derived from interval hypergraphs, characterizing when these posets form various types of lattices, including distributive and semidistributive ones.

Contribution

It provides a characterization of interval hypergraphs whose associated hypergraphic posets form specific lattice structures, advancing understanding of their combinatorial properties.

Findings

Identifies conditions under which $P_\mathbb{I}$ is a lattice.

Determines when $P_\mathbb{I}$ is distributive or semidistributive.

Classifies $P_\mathbb{I}$ as a lattice quotient of the weak order.

Abstract

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.