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In addition to inherent computational challenges, the absence of a canonicalmethod for quantifying `persistence' in multi-parameter persistent homologyremains a hurdle in its application.  One of the best known quantifications of persistence for multi-parameterpersistent homology is the rank invariant, which has recently evolved into thegeneralized rank invariant (GRI) by naturally extending its domain. Thisextension enables us to quantify persistence across a broader range of regionsin the indexing poset compared to the rank invariant. However, the size of thedomain of the GRI is generally formidable, making it desirable to restrict itsdomain to a more manageable subset for computational purposes. The foremostquestions regarding such a restriction of the domain are: (1) How to restrict,if possible, the domain of the GRI without any loss of information? (2) Whencan we more compactly encode the GRI as a `persistence diagram'? (3) What isthe trade-off between computational efficiency and the discriminating power ofthe GRI as the amount of the restriction on the domain varies? (4) What proxiesexist for persistence diagrams in the multi-parameter setting that can bederived from the GRI? To address the first three questions, we generalize andaxiomatize the classic fundamental lemma of persistent homology via the notionof M\"obius invertibility of the GRI which we propose. This extension alsocontextualizes known results regarding the (generalized) rank invariant withinthe classical theory of M\"obius inversion. We conduct a comprehensivecomparison between M\"obius invertibility and other existing concepts relatedto the structural simplicity of persistence modules.  We address the fourth question through the notion of motivic invariants. Wedemonstrate that many invariants from the literature can be both derived fromthe GRI and recast as motivic invariants.

The Generalized Rank Invariant: M\"obius invertibility, Discriminating Power, and Connection to Other Invariants