Generators of modules in tropical geometry

A module $M$ over the tropical semifield $T$ is analogous to a module over a field. We assume that $M$ is straight reflexive, and define the dimension of $M$ to the number of elements of a basis. We study the dimension of a straight reflexive submodule $N \subset M$. Also we find an enough condition to the reflexivity. This result has an application to polytopes in a tropical projective space, and also to a Riemann-Roch theorem for tropical curves.