On equations and first-order theory of one-relator monoids
| Autor: | Garreta, Albert, Gray, Robert D. |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: v2: The paper has been restructured and retitled, and helpful suggestions made by an anonymous referee have been implemented |
| Databáze: | arXiv |
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