Completeness and stability of the class of injective $S$-acts

We deal with questions concerning the completeness and stability of a class of injective acts and a class of weakly injective acts over a monoid $S$. The concepts of an injective $S$-act and of a weakly injective $S$-act are analogs of the concept of an injective module. In the theory of modules, the corresponding notions of injectivities in accordance with Baer's criterion coincide. Also we will look into completeness and stability of a class of principally weakly injective $S$-acts and a class of fg-weakly injective $S$-acts, which are analogs of $p$-injective modules and finitely injective modules.