Maps $f$ defined on the interior of the standard non-negative cone $K$ in ${ℝ}^N$ which are both homogeneous of degree $1$ and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson’s part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that the extension must have at least...

We are interested here in the reachability and controllability problems for DEDS in the max-algebra. Contrary to the situation in linear systems theory, where controllability (resp observability) refers to a (linear) subspace, these properties are essentially discrete in the $max$-linear dynamic system. We show that these problems, which consist in solving a $max$-linear equation lead to an eigenvector problem in the $min$-algebra. More precisely, we show that, given a $max$-linear system, then, for every natural...