辫状群和阿尔丁群的成员问题

Robert D. Gray, Carl-Fredrik Nyberg-Brodda
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引用次数: 0

摘要

我们研究了辫状群和阿尔丁群中的几个自然判定问题。我们将定义图中不存在某些禁止诱导子图的阿尔丁群归类为具有可判定子单体成员资格问题的阿尔丁群。此外,我们还对以下问题可解的 Artin 群进行了分类:有理子集成员资格问题、半群相交问题、固定目标子单体成员资格问题和有理同一性问题。在辫状群的情况下,我们的结果表明,在辫状群 $\mathbf{B}_n$ 中,当且仅当 $n \leq3$时,子集成员资格问题以及这些问题中的每一个都是可解的,这回答了 Potapov(2013)的一个未决问题。我们的结果还概括并扩展了 Lohrey & Steinberg(2008)的结果,他们将直角阿汀群归类为具有可解的子元组(及有理子集)成员资格问题的直角阿汀群。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Membership problems in braid groups and Artin groups
We study several natural decision problems in braid groups and Artin groups. We classify the Artin groups with decidable submonoid membership problem in terms of the non-existence of certain forbidden induced subgraphs of the defining graph. Furthermore, we also classify the Artin groups for which the following problems are decidable: the rational subset membership problem, semigroup intersection problem, fixed-target submonoid membership problem, and the rational identity problem. In the case of braid groups our results show that the submonoid membership problem, and each and every one of these problems, is decidable in the braid group $\mathbf{B}_n$ if and only if $n \leq 3$, which answers an open problem of Potapov (2013). Our results also generalize and extend results of Lohrey & Steinberg (2008) who classified right-angled Artin groups with decidable submonoid (and rational subset) membership problem.
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