Is decidability of the Submonoid Membership Problem closed under finite extensions?
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929351648346112 |
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| author | Shafrir, Doron |
| author_facet | Shafrir, Doron |
| contents | We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a finite index subgroup uniformly for all virtually nilpotent groups. We also provide evidence towards the existence of a group $G$ with a subgroup $H<G$ of index 2, such that the submonoid membership problem is decidable in $H$ but not in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Is decidability of the Submonoid Membership Problem closed under finite extensions? Shafrir, Doron Group Theory Formal Languages and Automata Theory We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a finite index subgroup uniformly for all virtually nilpotent groups. We also provide evidence towards the existence of a group $G$ with a subgroup $H<G$ of index 2, such that the submonoid membership problem is decidable in $H$ but not in $G$. |
| title | Is decidability of the Submonoid Membership Problem closed under finite extensions? |
| topic | Group Theory Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2405.12921 |