A family of simplicial complexes connected by simplicial maps and indexed by a finite poset $P$ is called a poset tower. Poset towers subsume multi-parameter filtrations, zigzag filtrations, and one-parameter simplicial towers, while allowing arbitrary finite posets and simplicial maps. The homology of a poset tower is a $P$-persistence module. To compute it globally over $P$, we consider the chain complex segment of $P$-persistence modules $C_{\ell-1}\xleftarrow{\partial_{\ell}}C_\ell \xleftarrow{\partial_{\ell+1}}C_{\ell+1}$ induced by the simplices of the tower. Unlike in one-critical multi-filtrations, the chain modules $C_\ell$ need not be projective and may have a complicated structure. We address the problem of replacing this segment by projective modules and $P$-graded matrices while preserving homology. The resulting projective implicit representation (PiRep) plays the role of the graded boundary-matrix representation in the classical persistence algorithm: it converts simplicial data into algebraic input on which persistent homology can be computed globally over $P$. In particular, a PiRep can be used as input to algorithms for computing minimal presentations of persistent homology. We give an efficient algorithm to compute a PiRep from a poset tower. It constructs degreewise minimal presentations and asymptotically minimal second terms of projective resolutions of the chain modules $C_\ell$, lifts the boundary maps $\partial_\ell$ to these resolutions, and assembles the resulting data into a PiRep using an additional correction term. The method is tailored to chain complexes induced by poset towers and computes the required algebraic data combinatorially, exploiting their special structure and avoiding general-purpose algebraic reduction. In the context of poset towers, it is fully general and can serve as a foundation for efficient algorithms on specific posets.

翻译：一族由单射映射连接、并以有限偏序集$P$索引的单纯复形称为偏序集塔。偏序集塔概括了多参数过滤、之字形过滤和单参数单纯塔，同时允许任意有限偏序集和单射映射。偏序集塔的同调是一个$P$-持久性模。为了在$P$上全局计算该同调，我们考虑由塔的单纯形诱导的$P$-持久性模的链复形片段$C_{\ell-1}\xleftarrow{\partial_{\ell}}C_\ell \xleftarrow{\partial_{\ell+1}}C_{\ell+1}$。与单临界多重过滤不同，链模$C_\ell$不一定是射影的，且可能具有复杂结构。我们解决了在保持同调的前提下，将该片段替换为射影模和$P\)-分次矩阵的问题。由此得到的射影隐式表示（PiRep）在经典持久性算法中扮演了分次边界矩阵表示的角色：它将单纯数据转化为代数输入，从而可在$P$上全局计算持久同调。特别地，PiRep可作为算法输入用于计算持久同调的最小表示。我们给出了一种从偏序集塔高效计算PiRep的算法。该算法逐度构造链模$C_\ell$射影分解的最小表示和渐近最小的第二项，将边界映射$\partial_\ell$提升到这些分解上，并通过附加修正项将所得数据组装成PiRep。该方法专为偏序集塔诱导的链复形设计，利用其特殊结构进行组合计算，避免了通用代数化简。在偏序集塔的背景下，该方法是完全通用的，可为特定偏序集上的高效算法奠定基础。