We 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.Comment: 24 pages, 5 Postscript figures, revised (v2

Abstrucf-A discrete-event system is a system whose behavior can be described by means of a set of time-consuming activities, performed according to a prescribed ordering. Events correspond to starting or ending some activity. An analogy between b e a r systems and a class of discrete-event systems is developed. Following this analogy, such discreteevent systems can be viewed as linear, in the sense of an appropriate algebra. The periodical behavior of closed discrete-event systems, i.e., involving a set of repeatedly performed activities, can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra. This problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph. The potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized; the case of a flowshop-like production process is analyzed in detail. L recommended by P. R. Kumar. Past Chairman of the Stochastic Control Manuscript

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In this paper, it is shown that a certain class of Petri nets called event graphs can be represented as linear "time-invariant" finite-dimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developped in a manner which is very analogous to that of conventional linear system theory. Part 2 of the paper is devoted to showing some preliminary basic developments in that direction. Indeed, there are several ways in which one can consider event graphs as linear systems: these ways correspond to approaches in the time domain, in the event domain and in a two-dimensional domain. In each of these approaches, a different algebra has to be used for models to remain linear. However, the common feature of these algebras is that they all fall into the axiomatic definition of "dioids". Therefore, Part 1 of the paper is devoted to a unified presentation of basic algebraic results on dioids.

We describe the specialization to max-plus algebra of Howard's policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the max-plus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time.