Uniform semimodular lattices and valuated matroids

TL;DR

This paper introduces uniform semimodular lattices as a new lattice-theoretic framework to characterize valuated matroids, extending classical matroid-lattice correspondences and connecting to tropical geometry and Euclidean buildings.

Contribution

It defines uniform semimodular lattices and proves a cryptomorphic equivalence with integer-valued valuated matroids, expanding the theoretical understanding of these structures.

Findings

Establishes a lattice-theoretic characterization of valuated matroids.

Provides a coordinate-free description of tropical linear spaces.

Connects valuated matroids with Euclidean buildings of type A.

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.