Amalgams of matroids, fibre products and tropical graph correspondences

We prove that the proper amalgam of matroids M₁ and M₂ along their common restriction N exists if and only if the tropical fibre product of Bergman fans B(M₁) ×B(N) B(M₂) is positive. We introduce tropical correspondences between Bergman fans as tropical subcycles in their product, similar to correspondences in algebraic geometry, and define a "graph correspondence" of the map of lattices. We prove that graph construction is a functor for the "covering" maps of lattices, exploiting a generalization of Bergman fan which we call a "Flag fan".