  Local conditions for the direct summands of a persistence module to belong to
a certain class of indecomposables have been proposed in the 2-parameter
setting, notably for the class of indecomposables called block modules, which
plays a prominent role in levelset persistence. Here we generalize the local
condition for decomposability into block modules to the n-parameter setting,
and prove a corresponding structure theorem. Our result holds in the generality
of pointwise finite-dimensional modules over finite products of arbitrary
totally ordered sets. Our proof extends the one by Botnan and Crawley-Boevey
from 2 to n parameters, which requires some crucial adaptations at places where
their proof is fundamentally tied to the 2-parameter setting.
