  Finite dynamical systems (FDSs) are commonly used to model systems with a
finite number of states that evolve deterministically and at discrete time
steps. Considered up to isomorphism, those correspond to functional graphs. As
such, FDSs have a sum and product operation, which correspond to the direct sum
and direct product of their respective graphs; the collection of FDSs endowed
with these operations then forms a semiring. The algebraic structure of the
product of FDSs is particularly interesting. For instance, an FDS can be
factorised if and only if it is composed of two sub-systems running in
parallel. In this work, we further the understanding of the factorisation,
division, and root finding problems for FDSs. Firstly, an FDS $A$ is
cancellative if one can divide by it unambiguously, i.e. $AX = AY$ implies $X =
Y$. We prove that an FDS $A$ is cancellative if and only if it has a fixpoint.
Secondly, we prove that if an FDS $A$ has a $k$-th root (i.e. $B$ such that
$B^k = A$), then it is unique. Thirdly, unlike integers, the monoid of FDS
product does not have unique factorisation into irreducibles. We instead
exhibit a large class of monoids of FDSs with unique factorisation. To obtain
our main results, we introduce the unrolling of an FDS, which can be viewed as
a space-time expansion of the system. This allows us to work with (possibly
infinite) trees, where the product is easier to handle than its counterpart for
FDSs.
