Pure Dimension and Projectivity of Tropical Polytopes

We give geometric and order-theoretic characterisations of those finitely generated convex sets over the tropical semiring which are projective modules. Specifically, we show that a finitely generated convex set is projective if and only if it has pure dimension equal to its generator dimension and dual dimension. We also give an order-theoretic description of projectivity in terms of sets which are both max-plus and min-plus closed. Our results yield information about the algebraic structure of tropical matrices under multiplication, including a geometric and order-theoretic understanding of idempotency and von Neumann regularity. A consequence is that many of the numerous notions of rank which are studied for tropical matrices coincide for von Neumann regular (and, in particular, idempotent) matrices.