# On equations and first-order theory of one-relator monoids

## TL;DR

This paper explores the logical and algebraic properties of certain one-relator monoids, showing that many have undecidable first-order theories by linking their equations to the open problem of word equations with length constraints.

## Contribution

It introduces a family of one-relator monoids where the decidability of word equations reduces to their Diophantine problem, revealing undecidability of their first-order theories.

## Key findings

- Many monoids in the family have undecidable positive AE-theory.
- Includes monoids with torsion where word equations' decidability is linked to Diophantine problem.
- Contrasts with hyperbolic groups, which have decidable Diophantine problems.

## Abstract

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 $\langle A\mid w=1\rangle$ 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 $\langle A\mid w^n = 1\rangle$ ($n>1$). 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.