  Multiparameter persistence is a natural extension of the well-known
persistent homology, which has attracted a lot of interest. However, there are
major theoretical obstacles preventing the full development of this promising
theory.
  In this paper we consider the interesting special case of multiparameter
persistence in zero dimensions which can be regarded as a form of
multiparameter clustering. In particular, we consider the multiparameter
persistence modules of the zero-dimensional homology of filtered topological
spaces when they are finitely generated. Under certain assumptions, we
characterize such modules and study their decompositions. In particular we
identify a natural class of representations that decompose and can be extended
back to form zero-dimensional multiparameter persistence modules.
  Our study of this set of representations concludes that despite the
restrictions, there are still infinitely many classes of indecomposables in
this set.
