Interval hypergraphic lattices

Abstract

For a hypergraph ℍ on [ ] , the hypergraphic poset ℍ is the transitive closure of the oriented skeleton of the hypergraphic polytope △ℍ (the Minkowski sum of the standard simplices △ for all ∈ℍ ). Hypergraphic posets include the weak order for the permutahedron (when ℍ is the complete graph on [ ] ) and the Tamari lattice for the associahedron (when ℍ is the set of all intervals of [ ] ), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of [ ] . We characterize the interval hypergraphs for which is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order.