  In the persistent homology of filtrations, the indecomposable decompositions
provide the persistence diagrams. However, in almost all cases of
multidimensional persistence, the classification of all indecomposable modules
is known to be a wild problem. One direction is to consider the subclass of
interval-decomposable persistence modules, which are direct sums of interval
representations. We introduce the definition of pre-interval representations, a
more natural algebraic definition, and study the relationships between
pre-interval, interval, and indecomposable thin representations. We show that
over the ``equioriented'' commutative $2$D grid, these concepts are equivalent.
Moreover, we provide a criterion for determining whether or not an $n$D
persistence module is interval/pre-interval/thin-decomposable without having to
explicitly compute decompositions. For $2$D persistence modules, we provide an
algorithm together with a worst-case complexity analysis that uses the total
number of intervals in an equioriented commutative $2$D grid. We also propose
several heuristics to speed up the computation.
