AI 中文总结
该研究探讨有限Coxeter群绝对序中拟Coxeter元下方区间的组合学，证明其EL-可壳性（E₇、E₈型可能例外），计算ζ多项式，证明Williams关于Coxeter元的猜想，还发现相关区间幺半群的可消去性反例，否定了Baumeister等人的问题。
AI 中文摘要
我们研究有限Coxeter群绝对序中拟Coxeter元下方区间的组合学。特别地，我们证明这些区间是EL-可壳的，仅E₇和E₈型的拟Coxeter元可能例外，并计算了它们的ζ多项式。我们还证明：若反射的全序在所有秩二子区间上均能产生EL-标号，则该全序可产生EL-标号，这证明了Williams关于Coxeter元的一个猜想。为获得这些结果，我们先对无限族证明相关结论，再通过计算机处理例外型。此外，我们研究了相关的区间幺半群，发现对所有Dₙ（n≥5）、E₆、E₇、E₈型的真拟Coxeter元，存在可消去性的反例；特别地，这些幺半群无法嵌入对应的区间群中。这对Baumeister、Holt、Neaime和Rees提出的问题给出了否定回答。
英文摘要
We study the combinatorics of the intervals below quasi-Coxeter elements in the absolute order of finite Coxeter groups. In particular, we prove that these intervals are EL-shellable, except possibly for quasi-Coxeter elements of types $E_7$ and $E_8$, and we compute their zeta polynomials. We also show that a total order of the reflections yields an EL-labeling if it does so on all rank-two subintervals, which proves a conjecture of Williams for Coxeter elements. In order to obtain these results, we prove them for the infinite families, and we treat the exceptional types by computer. In addition, we study the related interval monoids, and we find a counterexample to cancellativity for every proper quasi-Coxeter element of type $D_n$ ($n\ge5$), $E_6$, $E_7$, $E_8$; in particular, these monoids do not embed in the corresponding interval groups. This gives a negative answer to a question posed by Baumeister, Holt, Neaime, and Rees.
