  We consider functions $f(v)=\min_{A\in K}{Av}$ and $g(v)=\max_{A\in K}{Av}$,
where $K$ is a finite set of nonnegative matrices and by "min" and "max" we
mean coordinate-wise minimum and maximum. We transfer known results about
properties of $g$ to $f$. In particular we show existence of nonnegative
generalized eigenvectors for $f$, give necessary and sufficient conditions for
existence of strictly positive eigenvector for $f$, study dynamics of $f$ on
the positive cone. We show the existence and construct matrices $A$ and $B$,
possibly not in $K$, such that $f^n(v)\sim A^nv$ and $g^n(v)\sim B^nv$ for any
strictly positive vector $v$.
