Uniform semimodular lattices and valuated matroids Hiroshi HIRAI Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan.
[email protected] March 4, 2019 Abstract In this paper, we present a lattice-theoretic characterization for valuated ma- troids, which is an extension of the well-known cryptomorphic equivalence be- tween 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 ma- troids and uniform semimodular lattices. Our result includes a coordinate-free lattice-theoretic characterization of integer points in tropical linear spaces, incor- porates the Dress-Terhalle completion process of valuated matroids, and estab- lishes a smooth connection with Euclidean buildings of type A. Keywords: Valuated matroid, uniform semimodular lattice, geometric lattice, trop- ical linear space, tight span, Euclidean building. 1 Introduction Matroids can be characterized by various cryptomorphically equivalent axioms; see e.g., [1]. Among them, a lattice-theoretic characterization by Birkhoff [4] is well-known: The lattice of flats of any matroid is a geometric lattice (= atomistic semimodular lattice), and any geometric lattice gives rise to a simple matroid. The goal of the present article is to extend this classical equivalence to valuated arXiv:1802.08809v2 [math.CO] 1 Mar 2019 matroids (Dress and Wenzel [8, 9]). Valuated matroid is a quantitative generalization of matroid, which abstracts linear dependence structures of vector spaces over a field with a non-Archimedean valuation. A valuated matroid is defined as a function on the base family of a matroid satisfying a certain exchange axiom that originates from the Grassmann-Pl¨ucker identity.