Recently the poset of ranked cactuses $(\mathfrak{P}(X),\preceq)$ was introduced. For a finite set $X$, this poset consists of a set $\mathfrak{P}(X)$ of certain collections of ordered pairs of subsets of $X$ together with an ordering $\preceq$ that is similar to the refinement ordering of partitions of a finite set. In addition, the maximal chains in this poset correspond to binary ranked cactuses, a fact which can be used to construct the so-called space of equidistant cactuses. In this paper, we show that the poset of ranked cactuses is EL-shellable. As a consequence we also show that the proper part of the link of the origin of the space of equidistant cactuses has the homotopy type of a wedge of spheres.

翻译：最近引入了分级仙人掌偏序集 $(\mathfrak{P}(X),\preceq)$。对于有限集合 $X$，该偏序集由集合 $\mathfrak{P}(X)$（包含某些由 $X$ 的子集构成的有序对组）连同一种类似有限集合划分细化的序关系 $\preceq$ 组成。此外，该偏序集中的极大链对应二元分级仙人掌，这一事实可用于构造所谓的等距仙人掌空间。本文证明了分级仙人掌偏序集具有 EL-可壳性。作为推论，我们还证明了等距仙人掌空间原点的联结的真部分具有球面楔积的同伦型。