2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66062We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.24 pages, 5 Postscript figures, revised (v2)Functional AnalysisOptimization and Control46A20 (Primary) 06F07, 46A55 (Secondary)text