We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.

中文翻译：

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关于半群直积的可分性

我们研究了与残差有限性相关的四种有限性条件：完全可分性、强子半群可分性、弱子半群可分性和单基因子半群可分性。对于这些特性中的每一个，我们检查在哪些条件下该特性在直接产品下得以保留。我们还考虑是否有任何属性由直接乘积中的因子继承。我们给出了有限半群的充要条件，以在直积中保持强子半群可分性和单基因子半群可分性的性质。