2018/02/24 by Hiroshi Hirai, Hirai, Hiroshi
Computer Science · Mathematics · #05B35 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1802.08809
openalex publication_date 2018/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.