In this paper we prove several results regarding decidability of the membership problem for certain submonoids in amalgamated free products and HNN extensions of groups. These general results are then applied to solve the prefix membership problem for a number of classes of one-relator groups which are low in the Magnus-Moldavanski\ui hierarchy. Since the prefix membership problem for one-relator groups is intimately related to the word problem for one-relator special inverse monoids in the $E$-unitary case (as discovered in 2001 by Ivanov, Margolis and Meakin), these results yield solutions of the word problem for several new classes of one-relator special inverse monoids. In establishing these results, we introduce a new theory of conservative factorisations of words which provides a link between the prefix membership problem of a one-relator group and the group of units of the corresponding one-relator special inverse monoid. Finally, we exhibit the first example of a one-relator group, defined by a reduced relator word, that has an undecidable prefix membership problem.

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单相关组前缀成员问题的新结果

在本文中，我们证明了一些关于合并自由产品和组的 HNN 扩展中某些子单体的隶属度问题的可判定性的结果。然后将这些一般结果应用于解决 Magnus-Moldavanski\ui 层次结构中较低的单相关组的许多类的前缀成员资格问题。由于单关系群的前缀隶属关系问题与 $E$-酉情形中单关系特殊逆幺半群的词问题密切相关（如 Ivanov、Margolis 和 Meakin 于 2001 年发现的），这些结果产生了以下解：几个新类别的单相关特殊逆幺半群的词问题。在建立这些结果时，我们引入了一种新的单词保守因式分解理论，它提供了一个单关系群的前缀成员问题与相应单关系特殊逆幺半群的单位群之间的联系。最后，我们展示了一个单关系组的第一个例子，由一个简化的关系词定义，它有一个不可判定的前缀成员问题。