  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.
