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The Representation Theorem by Zomorodian and Carlsson has been the startingpoint of the study of persistent homology under the lens of algebraicrepresentation theory. In this work, we give a more accurate statement of theoriginal 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 theRepresentation Theorem of multidimensional persistence as a special case.

The Representation Theorem of Persistent Homology Revisited and Generalized