  In this note, we extend results about unique $n^{\textrm{th}}$ roots and
cancellation of finite disconnected graphs with respect to the Cartesian, the
strong and the direct product, to the rooted hierarchical products, and to a
modified lexicographic product. We show that these results also hold for graphs
with countably many finite connected components, as long as every connected
component appears only finitely often (up to isomorphism). The proofs are via
monoid algebras and generalized power series rings.
