Structural and categorical description of classes of abelian groups

The work presents results concerning the self-smallness and relative smallness properties of products in the category of abelian groups and in Ab5-categories. Criteria of relative smallness of abelian groups and a characterization of self-small products of finitely generated abelian groups are also given. A decomposition theory of UD-categories and in consequence a unified theory of decomposition in categories of S-acts is developed with applications on (auto)compactness proper- ties of S-acts. A structural description of compact objects in two categories of S-acts is provided. The existence of projective covers in categories S − Act and S − Act0 and the issue of perfectness of a monoid with zero are discussed.