Convex Geometry of Building Sets
Spencer Backman, Richard Danner
TL;DR
The paper generalizes building-set theory from lattices of flats to finite meet-semilattices by establishing three equivalent descriptions in this broader setting and showing that the associated building sets form a supersolvable convex geometry. It proves that the closed-sets of this convex geometry correspond to a closure operator induced by building sets and demonstrates that the collection $\mathbb{B}(\mathcal{L})$ forms an intersection lattice with a natural rank function, enabling a constructive generation of building sets. A key contribution is the restriction theory: under consistent embeddings $\mathcal{L}\hookrightarrow\mathcal{K}$, building sets and nested set complexes behave well, which yields corollaries linking Bergman fans of matroids to larger nestohedral fans. This unifies and extends results across oriented matroids, Bergman fans, and Hodge-theoretic studies of matroids, offering a common combinatorial framework and algorithmic avenues for constructing building sets.
Abstract
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinatorial abstraction of convexity. Supersolvable convex geometries and antimatroids appear in the study of poset closure operators, Coxeter groups, and matroid activities. We prove that the building sets on a finite meet-semilattice form a supersolvable convex geometry. As an application, we demonstrate that building sets and nested set complexes respect certain restrictions of finite meet-semilattices unifying and extending results of several authors.