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In this paper, the categorial property of compactness of an object, i. e.commuting of the corresponding $\Hom$ functor with coproducts, is studied incategories of $S$-acts and the corresponding structural properties of compact$S$-acts are shown. In order to establish a general context and to unify theapproach to both of the most important categories of $S$-acts, the notion of aconcrete category with unique decomposition of objects is introduced andstudied.

Connected objects in categories of $S$-acts