AI 中文总结
本文研究两生成元单关系逆幺半群的字问题，证明了其可判定与不可判定结果，给出首个不可判定字问题的E-酉两生成元单关系逆幺半群等例子，还给出其可判定字问题的充分条件。
AI 中文摘要
我们研究两生成元单关系逆幺半群中的字问题，证明了其可判定性与不可判定性结果。我们给出了首个形如 $\boldsymbol{\text{Inv}} \boldsymbol{\text{Inv} \rangle$ 的两生成元单关系逆幺半群例子，该例子具有不可判定的字问题，且其可选取为 E-酉的。为证明该结果，我们利用了Ivanov、Margolis与Meakin（2001）的研究中建立的单关系逆幺半群字问题与单关系子群的子幺半群成员问题之间的联系。我们还给出了首个具有简约定义关系的两生成元单关系子群例子，该例子具有不可判定的前缀成员问题。我们对具有可判定字问题的两生成元单关系逆幺半群的分类问题展开研究，为此证明了一个一般性结果：具有多项式增长单值群的自由-循环群何时具有可判定的子幺半群成员问题。具体而言，我们证明，具有多项式增长单值群的自由-循环群，其具有可判定的子幺半群成员问题当且仅当它具有可判定的有理子集成员问题，当且仅当定义自同构在外自同构群中具有有限阶。我们随后将该结果应用于，当两生成元单关系逆幺半群的极大群像为多项式增长的自由-循环群时，给出其具有可判定字问题的充分条件，且我们证明该条件可通过算法进行检验。相反，我们证明当该条件不满足时，该两生成元单关系逆幺半群可被提升为具有不可判定字问题且具有相同极大群像（与无限循环群的自由积）的例子。
英文摘要
We study the word problem in two-generator one-relator inverse monoids, proving both decidability and undecidability results. We give the first example of a two-generator one-relator inverse monoid of the form $\mathrm{Inv} \langle a,b \mid w=1 \rangle$ with an undecidable word problem. Furthermore our examples can be chosen to be $E$-unitary. To prove this result we exploit the connection between the word problem for one-relator inverse monoids and the submonoid membership problem for one-relator groups established in work of Ivanov, Margolis, and Meakin (2001). We also give the first example of a two-generator one-relator group, with a reduced word defining relator, with an undecidable prefix membership problem. We investigate the problem of classifying the two-generator one-relator inverse mononids with decidable word problem. For this we prove a general result classifying when a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem. Specifically we prove that a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem if and only if it has decidable rational subset membership problem if and only if the defining automorphism has finite order in the outer automorphism group. We then apply that result to give a sufficient condition for a two-generator one-relator inverse monoid to have decidable word problem in the case its maximal group image is free-by-cyclic of polynomial growth, and we show that that condition can be algorithmically checked. Conversely, we show that when this condition is not satisfied then the two-generator one-relator inverse monoid can be lifted to an example with undecidable word problem and with the same maximal group image (up to taking a free product with the infinite cyclic group).
Comments21 pages
