Lattice characterization of cyclic interval hypergraphic posets

TL;DR

This paper characterizes when hypergraphic posets derived from cyclic interval hypergraphs form lattices, extending previous results for interval and complete cyclic interval hypergraphs.

Contribution

It provides a complete lattice characterization for cyclic interval hypergraphs, broadening understanding of hypergraphic posets and their lattice properties.

Findings

Characterization of lattice conditions for cyclic interval hypergraphs

Extension of previous results for interval and complete cyclic cases

Connection between hypergraph structure and lattice properties

Abstract

Hypergraphic polytopes $\Delta_{\mathbb{H}}$ arise as Minkowski sums of simplices indexed by the hyperedges of a hypergraph $\mathbb{H}$. Orienting the $1$-skeleton of such a polytope by a certain generic linear functional gives rise to the hypergraphic poset $P_{\mathbb{H}}$. Hypergraphic posets include the weak order for the permutahedron and the Tamari lattice for the associahedron. This motivates the problem of determining when $P_{\mathbb{H}}$ is a lattice. In this paper, we give a complete lattice characterization for cyclic interval hypergraphs, extending the result of Bergeron and Pilaud for interval hypergraphs, and the result of Adenbaum et al. for the complete cyclic interval hypergraph.