AI 中文总结
本文通过高维重写和 $N$-tableau 模型，为有限秩 stylic 幺半群构造了有限相干表示，并利用同伦约简将其转化为标准字母生成元的有效有限同伦基。
AI 中文摘要
我们从高维重写的视角研究 stylic 幺半群。相干表示不仅记录幺半群的定义关系，还记录这些关系之间的关系。利用 $N$-tableau 模型，我们为每个有限秩的 stylic 幺半群构造了一个有限的相干表示。我们从已知的有限收敛行表示出发。通过为每个临界分支附加一个生成合流，Squier 相干定理给出了一个有限的相干扩展。然后，我们通过两次适应于 $N$-tableaux 行组合的连续同伦约简，简化所得的相干表示。最后，相干 Tietze 变换和进一步的同伦约简将构造从辅助行生成元转移到标准字母生成元。这为标准字母表示提供了一个可有效计算的有限同伦基。该构造在秩 $2$ 中详细说明。
英文摘要
We study stylic monoids from the viewpoint of higher-dimensional rewriting. A coherent presentation records not only the defining relations of a monoid, but also the relations among them. Using the $N$-tableau model, we construct a finite coherent presentation for every stylic monoid of finite rank. We start from the known finite convergent row presentation. By adjoining one generating confluence for each critical branching, Squier's coherence theorem yields a finite coherent extension. We then simplify the resulted coherent presentation by two successive homotopical reductions adapted to the row combinatorics of $N$-tableaux. Finally, coherent Tietze transformations and a further homotopical reduction transfer the construction from the auxiliary row generators to the standard letter generators. This gives an effectively computable finite homotopy basis for the standard letter presentation. The construction is illustrated in detail in rank $2$.
