On interval decomposability of 2D persistence modules

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