Rational Semimodules over the Max-plus Semiring and Geometric Approach of Discrete Event Systems

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring. We say that a subsemimodule of the free semimodule Sn over a semiring S is rational if it has a generating family that is a rational subset of Sn, Sn 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). Rational semimodules are a tool to extend the geometric approach of linear control to discrete event systems. In particular, we show that the reachable and observable spaces of max-plus linear dynamical systems are rational.