标题： 由序嵌入定义的持久性模块的不变量 显示英文标题

标题： Invariants of persistence modules defined by order-embeddings

摘要： 拓扑数据分析的主要目标之一是研究持久性模块的离散不变量，特别是在处理多参数持久性模块时。 在许多情况下，针对这些非全序偏序集 $P$ 所研究的不变量可以从将给定模块限制到一个子偏序集 $X$ 的 $P$ 上获得，该子偏序集是全序的（或更一般地，是有限表示类型），然后在 $X$ 上计算条形码（或一般的直和分解）。 我们在这篇论文中考虑了表示有限子偏序集$X$到$P$的一般保序嵌入，并系统地研究了通过将给定的$P$-模$M$限制到$X$并分解为其不可约和式所得的不变量。 从$\mathrm{mod}\ P$到$\mathrm{mod}\ X$的限制函子已被广泛研究，已知它是正合的，并且具有左伴随和右伴随函子，称为诱导函子和余诱导函子。 这使我们能够获得新的同调见解，并重新解释以前的结果。 我们还使用这种方法来确定这些不变量的像的基，从而推广了文献中与稳定性结果相关的符号条形码的概念。 结果表明，仅考虑将一个固定的偏序集$X$嵌入到偏序集$P$中，并研究从$X$获得的所有不可分解结构，会引入很多冗余。因此，我们还研究了多个大小递增的偏序集的迭代嵌入，同时仅关注一些不可分解结构（这些结构之前未通过较小偏序集的嵌入获得）。 显示英文摘要

摘要： One of the main objectives of topological data analysis is the study of discrete invariants for persistence modules, in particular when dealing with multiparameter persistence modules. In many cases, the invariants studied for these non-totally ordered posets $P$ can be obtained from restricting a given module to a subposet $X$ of $P$ that is totally ordered (or more generally, of finite representation type), and then computing the barcode (or the general direct sum decomposition) over $X$. We consider in this paper general order-preserving embeddings of representation-finite subposets $X$ into $P$ and study systematically the invariants obtained by decomposing the restriction of a given $P$-module $M$ to $X$ into its indecomposable summands. The restriction functor from $\mathrm{mod}\ P$ to $\mathrm{mod}\ X$ is well-studied, and it is known to be exact and admits both left and right adjoint functors, known as induction and co-induction functors. This allows us to obtain new homological insights, and also to re-interpret previous results. We use this approach also to determine bases of the image of these invariants, thus generalizing the concept of signed barcodes which is considered in the literature in relation to stability results. It turns out that considering only order-embeddings of one fixed poset $X$ into the poset $P$, and studying the set of all indecomposables obtained from $X$ introduces a lot of redundancy. We therefore also study iterated embeddings of several posets of increasing sizes, while limiting attention to only some indecomposables (that have not been obtained from embedding of smaller posets previously).