Uniform semimodular lattices and valuated matroids

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:165

Article

Hirai, Hiroshi¹ 

[1] Univ Tokyo, Grad Sch Informat Sci & Technol, Dept Math Informat, Tokyo 1138656, Japan

关键词: Valuated matroid;    Uniform semimodular lattice;    Geometric lattice;    Tropical linear space;    Tight span;    Euclidean building;   

DOI  :  10.1016/j.jcta.2019.02.013

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free