Unbounded matroids

TL;DR

This work introduces unbounded matroids (U-matroids) by placing matroid rank data on a distributive lattice 𝒟, yielding base polyhedra that are extended generalized permutahedra with 0/1 vertices. A central contribution is the generous extension, a canonical process that embeds every U-matroid into a matroid by producing the largest contained matroid base polytope, and its geometric counterpart as the largest sheared polytope within a given base polyhedron. The pseudo-independence complex Δ(U) is shown to be shellable, generalizing Björner’s and Gale’s matroid criteria to U-matroids via maximal chains in the characteristic lattice, and duality is established in parallel to the classical theory. The framework connects to subspace arrangements: generous extension corresponds to the multisymmetric lift, providing a unifying view of poset matroids and multisymmetric matroids, and offering concrete interpretations for representability and geometric interpretations. Overall, the paper lays a robust polyhedral/combinatorial foundation for unbounded matroids and opens avenues for further axiomatizations and applications in arrangement theory and beyond.

Abstract

A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a boolean lattice, satisfying the unit increase property. We define a more general class of unbounded matroids, or U-matroids, by replacing the boolean lattice with an arbitrary distributive lattice. U-matroids thus serve as a combinatorial model for polyhedra that satisfy the vertex and edge conditions of matroid base polytopes, but may be unbounded. Like polymatroids, U-matroids generalize matroids and arise as a special case of submodular systems. We prove that every U-matroid admits a canonical largest extension to a matroid, which we call the generous extension; the analogous geometric statement is that every U-matroid base polyhedron contains a unique largest matroid base polytope. We show that the supports of vertices of a U-matroid base polyhedron span a shellable simplicial complex, and we characterize U-matroid basis systems in terms of shelling orders, generalizing Björner's and Gale's criteria for a simplicial complex to be a matroid independence complex. Finally, we present an application of our theory to subspace arrangements and show that the generous extension has a natural geometric interpretation in this setting.

Figures (5)

• Figure 1: Matroids, matroid base polytopes, and their generalizations. Horizontal lines are bijections, the rest inclusions. The sides of the cube have these meanings: left/right = combinatorial/geometric; bottom/top = integral/real; front/back = bounded/possibly unbounded.
• Figure 2: Two U-polyhedra with the same characteristic poset but different rank functions.
• Figure 3: The stalactite.
• Figure 4: "Cutting off" a polyhedron $\mathfrak p$ with a hyperplane $\ell(x)=a$ to produce a polytope $\mathfrak q$.
• Figure 5: Combinatorial models of subspace arrangements.

Theorems & Definitions (80)

• Definition 2.1: Submodular systems
• Remark 2.2
• Definition 2.3
• Theorem 2.4: Vertices of base polyhedra of submodular systems
• Proposition 2.5
• proof
• Theorem 2.6: Recession cones of base polyhedra of submodular systems
• Definition 3.1
• Definition 3.2
• Theorem 3.3