We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space $M$, rooted subsets relate the clustering behavior of the points of $M$ with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, M\'emoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

翻译：我们研究零维持久模块的分解，将其视为通过集合因子化的向量空间范畴中的函子。并非直接在向量空间层面处理，而是退一步，首先在集合层面研究分解问题。这一方法使我们能够定义根子集的组合概念。在过滤度量空间 $M$ 的情形下，根子集将 $M$ 中点的聚类行为与相关持久模块的分解联系起来。特别地，我们可以快速识别此类分解中的区间。此外，根子集可理解为长者规则（elder rule）的推广，并与Cai、Kim、Mémoli及Wang提出的恒定征服者（constant conqueror）概念相关。作为应用，我们给出了欧几里得空间中密度-Rips过滤的零维持久模块分解中预期区间数量的下界：在极限条件下且非常一般的情况下，可预期至少有25%的不可分解和项为区间模块。