Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup consisting of all products of n factors. Using this we show that a finite semigroup is Morita equivalent to its largest factorisable subsemigroup. It follows that two finite semigroups are Morita equivalent if and only if their Cauchy completions are equivalent categories. Since these categories are finite, the problem of Morita equivalence of finite semigroups is decidable.

中文翻译：

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有限半群的 Morita 等价

如果两个半群被称为 Morita 等价的，如果它们的企业权利行为的范畴是等价的。我们证明每个半群都是 Morita 等价于它的由 n 个因子的所有乘积组成的子半群。使用这个我们证明有限半群是 Morita 等价于它的最大可分解子半群。因此，两个有限半群是 Morita 等价的当且仅当它们的柯西完备是等价范畴。由于这些范畴是有限的，有限半群的 Morita 等价问题是可判定的。