We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz–Krieger $$C^*$$ C ∗ -algebras of singly aligned categories include the tight $$C^*$$ C ∗ -algebras of all countable inverse semigroups, up to Morita equivalence.

中文翻译：

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关于之字形映射和逆半群的路径范畴

我们研究了一个允许唯一最大幂等项的逆半群的路径范畴，并给出了由左抵消类别上的锯齿形映射产生的逆半群的抽象特征。作为应用，我们表明每个逆半群都是 Morita 等价于锯齿形映射的逆半群，因此 Cuntz–Krieger 的类 $$C^*$$ C ∗ - 单对齐类别的代数包括紧 $$C^*$ $ C ∗ -所有可数逆半群的代数，直到 Morita 等价。