AI 中文总结
该研究证明非全序有限偏序集索引的多持续模的结构定理，将厚度与凝聚性扩展到胞层，利用所得定理得到几何凝聚模的区间分解，并提出层弹性的双参数持续构造。
AI 中文摘要
我们证明了由非全序有限偏序集索引的多持续模的结构定理。具体而言，我们考虑有限集合非空子集构成的偏序集的对偶上的逐点有限维模，并通过态射给出此类模可分解为区间模直和的充分条件。一般情形下，区间直和项与重数由有限多个指标处的维数明确确定。尽管这些假设在代数上看似有局限性，我们证明它们在单纯复形上的胞层数据鲁棒性理论中有自然的几何起源，该理论研究代数信息、相容性约束与上同调障碍在结构失效下的持续性。我们将厚度与凝聚性从单纯上同调扩展到胞层：厚度检测上同调类对高维支集的依赖，凝聚性捕捉高阶邻接对上同调特征的影响。我们利用抽象结构定理得到所得几何凝聚模的区间分解。最后，我们引入层弹性的双参数持续构造，追踪拓扑退化过程中整体截面与上同调障碍是否在厚或凝聚子结构上保持可检测。
英文摘要
We prove structure theorems for multipersistence modules indexed by finite posets that are not totally ordered. Specifically, we consider pointwise finite-dimensional modules over the opposite of the poset of non-empty subsets of a finite set, and give sufficient conditions, expressed through transition morphisms, for such modules to split as direct sums of interval modules. In the general case, the interval summands and multiplicities are explicitly determined by dimensions at finitely many indices. Although the assumptions may look algebraically restrictive, we show that they have a natural geometric origin in a robustness theory of cellular sheaf data over simplicial complexes, where one studies how algebraic information, compatibility constraints, and cohomological obstructions persist under structural failures. We extend thickness and cohesion from simplicial cohomology to cellular sheaves: thickness detects the dependence of cohomology classes on high-dimensional support, while cohesion captures the influence of higher-order adjacencies on the cohomological features. We leverage our abstract structure theorems to obtain interval decompositions for the resulting geometric cohesion modules. Finally, we introduce biparameter persistence constructions for sheaf resilience, tracking whether global sections and cohomological obstructions remain detectable on thick or cohesive substructures during topological degradation.
