Interval hypergraphic lattices

TL;DR

This work provides a complete lattice-theoretic taxonomy for interval hypergraphic posets P_I arising from interval hypergraphs. It proves that P_I is a lattice exactly when I is closed under intersection, and that distributivity and semidistributivity correspond to stronger interval-closure conditions, with explicit characterizations and join-irreducible descriptions. It further analyzes Schröder (laminar) interval hypergraphs, yielding distributive lattices and factorized generating polynomials, and identifies how quotients of the weak order/Tamari lattice arise from subinterval-closure properties. The results connect the geometry of interval hypergraphic polytopes (deformations of the associahedron) to rich combinatorial structures (Tamari, Pitman–Stanley, and Schröder families) and provide concrete tools for understanding the poset and lattice structure via sources, flips, and canonical join representations.

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.

Figures (6)

• Figure 1: The polytope $\Delta_{\mathbb H}$ for $\mathbb H=\{ 1, 2, 3, 4, 123, 134 \}$ has seven vertices corresponding to the acyclic orientations of $\mathbb H$ and eleven (oriented) edges corresponding to the increasing flips between these orientations. The poset $P_\mathbb H$ is the transitive closure of the increasing flip graph.
• Figure 2: The poset morphism $\mathcal{O}$ from the weak order (left) to the hypergraphical poset $P_\mathbb H$ (right), for $\mathbb H=\{ 1, 2, 3, 4, 123, 134 \}$. The fibers of $\mathcal{O}$ appear as blue bubbles on the weak order, labeled by their acyclic orientation of $\mathbb H$ (left).
• Figure 3: The interval hypergraphical polytope $\triangle_\mathbb I$ (top left), the fibers of the corresponding map $\mathcal{O}$ (bottom left), and the interval hypergraphical poset $P_\mathbb I$ (right), for $\mathbb I = \{ 1, 2, 3, 4,123, 23, 234, 1234 \}$.
• Figure 4: Two interval hypergraphic posets which are not lattices.
• Figure 5: The Tamari lattice (semidistributive lattice, but not distributive).

Theorems & Definitions (159)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Example 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Example 2.5
• Definition 2.6