论文标题
亚森伯格集团有足够大的有限直接功率的下monoid成员问题的不确定性
Undecidability of the submonoid membership problem for a sufficiently large finite direct power of the Heisenberg group
论文作者
论文摘要
有限生成的组$ g $的下monoid成员资格问题是决策问题，在给定有限生成的$ m $ $ g $的$ g $和组元素$ g $的情况下，询问是否$ g \ in m $。在本文中，我们证明，对于Heisenberg Group $ \ Mathbb {H} $的足够大的直接功率$ \ mathbb {h}^n $，存在有限生成的sibonoid $ m $，其成员问题是算法上不可固定的。因此，关于Lohrey M. Lohrey和B. Steinberg的问题的答案是关于存在有限生成的Nilpotent群体的问题，具有无法解决的下monoid成员资格问题。它还回答了T. Colcombet，J。Ouaknine，P。Semukhin和J. Worrell的问题，介绍了Heisenberg Group的直接权力中存在这样的群体的问题。该结果意味着在任何自由的nilpotent $ n_ {k，c} $中存在类似的submonoid，该$ n_ {k，c} $ class $ c \ geq 2 $的足够大等级$ k $。证据是基于希尔伯特（Hilbert）的第10个问题的不可证明的，以及对尼尔替尼组中二芬太汀方程的解释。
The submonoid membership problem for a finitely generated group $G$ is the decision problem, where for a given finitely generated submonoid $M$ of $G$ and a group element $g$ it is asked whether $g \in M$. In this paper, we prove that for a sufficiently large direct power $\mathbb{H}^n$ of the Heisenberg group $\mathbb{H}$, there exists a finitely generated submonoid $M$ whose membership problem is algorithmically unsolvable. Thus, an answer is given to the question of M. Lohrey and B. Steinberg about the existence of a finitely generated nilpotent group with an unsolvable submonoid membership problem. It also answers the question of T. Colcombet, J. Ouaknine, P. Semukhin and J. Worrell about the existence of such a group in the class of direct powers of the Heisenberg group. This result implies the existence of a similar submonoid in any free nilpotent group $N_{k,c}$ of sufficiently large rank $k$ of the class $c\geq 2$. The proofs are based on the undecidability of Hilbert's 10th problem and interpretation of Diophantine equations in nilpotent groups.
