  We investigate the correspondence between generalized persistence modules and
graded modules in the case the indexing set has a monoid action. We introduce
the notion of an action category over a monoid graded ring. We show that the
category of additive functors from this category to the category of Abelian
groups is isomorphic to the category of modules graded over the set with a
monoid action, and to the category of unital modules over a certain smash
product. Furthermore, when the indexing set is a poset, we provide a new
characterization for a generalized persistence module being finitely presented.
