Is decidability of the Submonoid Membership Problem closed under finite extensions?
<|reference_start|>Is decidability of the Submonoid Membership Problem closed under finite extensions?: 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 $G$.<|reference_end|>
@article{shafrir2024is,
    title={Is decidability of the Submonoid Membership Problem closed under finite
  extensions?},
    author={Doron Shafrir},
    journal={arXiv preprint arXiv:2405.12921},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.12921},
    primaryClass={math.GR cs.FL}
}