  The Representation Theorem by Zomorodian and Carlsson has been the starting
point of the study of persistent homology under the lens of algebraic
representation theory. In this work, we give a more accurate statement of the
original theorem and provide a complete and self-contained proof. Furthermore,
we generalize the statement from the case of linear sequences of $R$-modules to
$R$-modules indexed over more general monoids. This generalization subsumes the
Representation Theorem of multidimensional persistence as a special case.
