Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module

A recent work by Lesnick and Wright proposed a visualisation of $2$D persistence modules by using their restrictions onto lines, giving a family of $1$D persistence modules. We give a constructive proof that any $1$D persistence module with finite support can be found as a restriction of some indecomposable $2$D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable $2$D persistence modules whose support has holes as well as an indecomposable $2$D persistence module containing all $1$D persistence modules with finite support as line restrictions. Finally, we also show that any finite-rectangle-decomposable $n$D persistence module can be found as a restriction of some indecomposable $(n+1)$D persistence module.