Uniform semimodular lattices and valuated matroids

Abstract

In this paper, we present a lattice-theoretic characterization for valuated matroids, which is an extension of the well-known cryptomorphic equivalence between matroids and geometric lattices ($=$ atomistic semimodular lattices). We introduce a class of semimodular lattices, called uniform semimodular lattices, and establish a cryptomorphic equivalence between integer-valued valuated matroids and uniform semimodular lattices. Our result includes a coordinate-free lattice-theoretic characterization of integer points in tropical linear spaces, incorporates the Dress-Terhalle completion process of valuated matroids, and establishes a smooth connection with Euclidean buildings of type A.

arXiv commentTo appear in Journal of Combinatorial Theory A

References in corpus (2)

• Computing the degree of determinants via discrete convex optimization on Euclidean buildings
• Computing Convex Hulls in the Affine Building of SL_d