The rank invariant (RI), one of the best known invariants of persistence modules $M$ over a given poset P, is defined as the map sending each comparable pair $p\leq q$ in P to the rank of the linear map $M(p\leq q)$. The recently introduced notion of generalized rank invariant (GRI) acquires more discriminating power than the RI at the expense of enlarging the domain of RI to the set Int(P) of intervals of P or to an even larger set. Given that the size of Int(P) can be much larger than that of the domain of the RI, restricting the domain of the GRI to smaller, more manageable subcollections $\mathcal{I}$ of Int(P) would be desirable to reduce the total cost of computing the GRI. This work studies the tension which exists between computational efficiency and strength when restricting the domain of the GRI to different choices of $\mathcal{I}$. In particular, we prove that the discriminating power of the GRI over restricted collections $\mathcal{I}$ strictly increases as $\mathcal{I}$ interpolates between the domain of RI and Int(P). Along the way, some well-known results regarding the RI or GRI from the literature are contextualized within the framework of the M\"obius inversion formula and we obtain a notion of generalize persistence diagram that does not require local finiteness of the indexing poset for persistence modules. Lastly, motivated by a recent finding that zigzag persistence can be used to compute the GRI, we pay a special attention to comparing the discriminating power of the GRI for persistence modules $M$ over $\mathbb{Z}^2$ with the so-called Zigzag-path-Indexed Barcode (ZIB), a map sending each zigzag path $\Gamma$ in $\mathbb{Z}^2$ to the barcode of the restriction of $M$ to $\Gamma$. Clarifying the connection between the GRI and the ZIB is potentially important to understand to what extent zigzag persistence algorithms can be exploited for computing the GRI.

翻译：秩不变量（RI）是定义在偏序集P上的持久模M的著名不变量之一，它通过将P中每个可比对p≤q映射为线性映射M(p≤q)的秩来定义。近年来引入的广义秩不变量（GRI）通过将RI的定义域扩展至P的区间集Int(P)或更广泛的集合，获得了比RI更强的判别能力。由于Int(P)的规模远大于RI的定义域，将GRI的定义域限制为Int(P)中更小、更易管理的子集类I，将有利于降低GRI的计算成本。本文研究了在将GRI的定义域限制为不同I的选择时，计算效率与判别强度之间的张力。特别地，我们证明当I从RI的定义域向Int(P)插值时，GRI在限制子类I上的判别能力严格递增。在此过程中，文献中关于RI或GRI的若干经典结果被置于Möbius反演公式框架下进行重新诠释，并由此获得了一个无需索引偏序集局部有限性的广义持久图概念。最后，受近期发现的"zigzag持久性可用于计算GRI"这一结果的启发，我们重点比较了定义在Z²上的持久模M的GRI与所谓Zigzag路径索引条形码（ZIB）之间的判别能力差异，其中ZIB将Z²中的每条zigzag路径Γ映射为M限制在Γ上的条形码。阐明GRI与ZIB之间的关系，对于理解如何利用zigzag持久性算法计算GRI具有重要意义。