Monoid algebras and graph products

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.