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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...