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

## TL;DR

This paper proves that every 1D persistence module with finite support can be realized as a restriction of an indecomposable 2D persistence module, expanding understanding of the structure and visualization of multidimensional persistence modules.

## Contribution

It provides a constructive proof that any finite-support 1D persistence module is a restriction of some indecomposable 2D persistence module, with new examples and generalizations to higher dimensions.

## Key findings

- Any 1D finite support persistence module is a restriction of an indecomposable 2D module.
- Existence of indecomposable 2D modules with support containing holes.
- Finite-rectangle-decomposable nD modules are restrictions of (n+1)D indecomposable modules.

## Abstract

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.