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We prove the following results: (1) There is a one-relator inverse monoid$\mathrm{Inv}\langle A\:|\:w=1 \rangle$ with undecidable word problem; and (2)There are one-relator groups with undecidable submonoid membership problem. Thefirst of these results answers a problem originally posed by Margolis, Meakinand Stephen in 1987.

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups