Quasivarieties and $q$-Compact Classes of Abelian Groups

For the purposes of algebraic geometry, we need to consider a category of Abelian $A$-groups, that is, those Abelian groups that contain as a subgroup the distinguished copy of an Abelian group $A$. Namely, we deal with the problem of describing $q$-compact classes within a given class of algebraic systems. This problem is solved first for classes of Abelian groups (without constants), and then for the case where a class of $A$-groups consists of the group $A$ itself. We also succeed in obtaining an adequate description of a system of axioms for $A-\operatorname{qvar}(B)$.