We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We establish an adjunction between the category ${\mathcal{P}}(S)$ of proper extensions of a restriction semigroup (or, in particular, an inverse semigroup) $S$ and a category ${\mathcal{A}}(S)$ of partial actions of $S$ subject to certain conditions going back to the work of O'Carroll. In the category ${\mathcal{A}}(S)$, we specify two isomorphic subcategories, one being reflective and the other one coreflective, each of which is equivalent to the category ${\mathcal{P}}(S)$.

中文翻译：

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两侧限制半群的部分作用和适当扩展

我们介绍并研究了双边限制半群的几类部分作用，它们概括了幺半群和逆半群的部分作用。我们证明了关于局部作用的两侧限制半群适当扩展的结构结果，推广了幺半群和逆半群的各自结果并升级了后者。我们在限制半群（或特别是逆半群）$S$ 的适当扩展的范畴 ${\mathcal{P}}(S)$ 和范畴 ${\mathcal{A}} 之间建立一个附属(S)$ $S$ 的部分行动受某些条件的约束，可以追溯到 O'Carroll 的工作。在类别 ${\mathcal{A}}(S)$ 中，我们指定了两个同构的子类别，一个是反射的，另一个是核心反射的，每个都等价于类别 ${\mathcal{P}}(S) $.