We prove the following results: (1) There is a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$ Inv ⟨ A | w = 1 ⟩ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg (J Algebra 320(2):728–755, 2008), we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$ Inv ⟨ A | w = 1 ⟩ with undecidable word problem. Furthermore, this method allows the construction of an E -unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis et al. (in: Semigroups and their applications, Reidel, Dordrecht, pp. 99–110, 1987).

中文翻译：

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通过单相关群的直角Artin子群的单相关逆幺半群的单词问题的不可判定性

我们证明以下结果： (1) 存在一个单关系逆幺半群 $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$ Inv ⟨ A | w = 1 ⟩ 具有不可判定的单词问题；(2) 存在不可判定的次幺半群成员问题的单相关群。这些结果中的第二个通过证明对于任何有限森林，关联的直角 Artin 群嵌入到一个单相关群中。将此与 Lohrey 和 Steinberg (J Algebra 320(2):728–755, 2008) 的结果相结合，我们用它来证明存在一个单相关群，其中包含一个固定的有限生成亚幺半群，其中隶属度问题是不可判定的. 为了证明（1），引入了一种新的构造，它使用单关系群和从（2）中成员资格不可判定的子幺半群来构造单关系逆幺半群 $$\mathrm {Inv}\langle A\,|\ , w=1 \rangle $$ Inv ⟨ A | w = 1 ⟩ 具有不可判定的单词问题。此外，该方法允许构造具有不可判定词问题的这种形式的 E-酉单相关逆幺半群。本文的结果回答了最初由 Margolis 等人提出的问题。(in: Semigroups and their applications, Reidel, Dordrecht, pp. 99–110, 1987)。