  It is shown that max-preserving maps (or join-morphisms) on the positive
orthant in Euclidean $n$-space endowed with the component-wise partial order
give rise to a semiring. This semiring admits a closure operation for maps that
generate stable dynamical systems. For these monotone maps, the closure is used
to define suitable notions of left and right eigenvectors that are
characterized by inequalities. Some explicit examples are given and
applications in the construction of Lyapunov functions are described.
