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The Diophantine problem for a monoid $M$ is the decision problem to decidewhether any given system of equations has a solution in $M$. In this note, wegive a simple example of a context-free, word-hyperbolic, finitely presented,special monoid $M$ with trivial group of units, and such that the Diophantineproblem is undecidable in $M$. This answers two questions asked by Garreta &Gray in 2019, and shows that the decidability of the Diophantine problem inhyperbolic groups, as proved by Dahmani & Guirardel, does not generalise toword-hyperbolic monoids.

A word-hyperbolic special monoid with undecidable Diophantine problem