  We introduce rational semimodules over semirings whose addition is
idempotent, like the max-plus semiring, in order to extend the geometric
approach of linear control to discrete event systems. We say that a
subsemimodule of the free semimodule S^n over a semiring S is rational if it
has a generating family that is a rational subset of S^n, S^n being thought of
as a monoid under the entrywise product. We show that for various semirings of
max-plus type whose elements are integers, rational semimodules are stable
under the natural algebraic operations (union, product, direct and inverse
image, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.
