We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$ . We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.

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关于特殊单位和一关系逆幺半群的群

我们研究单关系元和特殊逆幺半群的单位群。这些是由表示定义的逆幺半群，其中所有定义关系的形式为 $r=1$ 。我们开发了新的方法来寻找特殊反幺半群的单位群的表示，并应用这些方法来给出该群承认具有与幺半群相同数量的定义关系的表示的条件。特别是，我们的结果给出了单相关逆幺半群的单元群成为单相关群的充分条件。当满足这些条件时，这些结果给出了单关系幺半群的 Adjan 和特殊幺半群的 Makanin 的经典结果的逆半群理论类似物。相比之下，我们表明，一般来说，这些经典结果不适用于单关系子和特殊逆幺半群。特别地，我们证明存在一个单相关特殊逆幺半群，其单元群不是单相关群（相对于任何生成集），并且我们证明存在一个有限呈现的特殊逆幺半群，其群单位没有有限地呈现。