单关系群与单关系幺半群丢番图问题的不可判定性
Undecidability of the Diophantine problem for one-relator groups and one-relation monoids
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中文总结 AI 辅助
该研究证明，对所有 $d \geq 2$，Baumslag-Gersten群 $\operatorname{BG}(1,d)$ 与单关系幺半群 $\mathcal{M}_d$ 的丢番图问题不可判定，相关结果通过编码Rybalov的不可判定性成果得到。
中文摘要 AI 辅助
我们证明，一般而言，单关系群与单关系幺半群中的丢番图问题（即带系数的方程求解问题）是不可判定的。具体来说，对所有 $d \geq 2$，Baumslag-Gersten群 $\mathcal{B}_d = \operatorname{BG}(1,d)$ 和密切相关的单关系幺半群 $\mathcal{M}_d$ 的丢番图问题均不可判定，这两个结果均通过对Rybalov近期的一个不可判定性结果进行编码得到。
英文摘要
We prove that the Diophantine problem, i.e. the problem of solving equations with coefficients, in one-relator groups and one-relation monoids is undecidable in general. Specifically, we show that for all $d \geq 2$, the Diophantine problem for the Baumslag-Gersten group $\mathcal{B}_d = \operatorname{BG}(1,d)$ and for a closely related one-relation monoid $\mathcal{M}_d$ are both undecidable. Both results are obtained by an encoding of a recent undecidability result due to Rybalov.
