Closure operators and geometric modules of valuated matroids

We introduce a closure operator for valuated matroids and prove that it yields a cryptomorphic definition of valuated matroids. As an application, we introduce a class of modules over the tropical semifield (geometric modules) and prove that there is one-to-one correspondence between projective equivalence classes of simple valuated matroids and isomorphism classes of finitely generated geometric modules. We further illustrate how this correspondence can be lifted to simple infinite valuated matroids and geometric modules.