We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family $\mathcal{F}$ of one-relator monoids of the form $〈A|w=1〉$ where for each monoid M in $\mathcal{F}$, the longstanding open problem of decidability of word equations with length constraints reduces to the Diophantine problem (i.e. decidability of systems of equations) in M. We achieve this result by finding an interpretation in M of a free monoid, using only systems of equations together with length relations. It follows that each monoid in $\mathcal{F}$ has undecidable positive AE-theory, hence in particular it has undecidable first-order theory. The family $\mathcal{F}$ includes many one-relator monoids with torsion $〈A|w^n=1〉$ ($n>1$). In contrast, all one-relator groups with torsion are hyperbolic, and all hyperbolic groups are known to have decidable Diophantine problem. We further describe a different class of one-relator monoids with decidable Diophantine problem.

我们研究方程组和单相关幺半群的一阶理论。我们描述一个家庭$\mathcal{F}$ 形式的单关系幺半群 $〈\mathrm{一种}|瓦=1〉$其中对于每个半群中号在$\mathcal{F}$，具有长度约束的词方程的可判定性的长期开放问题简化为M 中的丢番图问题（即方程组的可判定性）。我们通过仅使用方程组和长度关系在M中找到自由幺半群的解释来实现这一结果。因此，每个幺半群在$\mathcal{F}$具有不可判定的正 AE 理论，因此特别是它具有不可判定的一阶理论。家庭$\mathcal{F}$ 包括许多具有扭转的单相关幺半群 $〈\mathrm{一种}|{瓦}^n=1〉$ ($n>1$）。相比之下，所有具有扭转的单相关群都是双曲的，并且已知所有双曲群都有可判定的丢番图问题。我们进一步描述了一类具有可判定丢番图问题的单相关幺半群。