Decomposition of pointwise finite-dimensional 1 persistence modules

We prove that, over an arbitrary field, pointwise finite-dimensional persistence modules indexed by S-1 decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. In the language of representation theory, this is a direct sum of string modules and band modules. Persistence modules indexed on S-1 have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on S-1 and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional S-1 persistence module is indecomposable if and only if it is a bar or Jordan cell. Along the way we classify the isomorphism classes of such indecomposable modules