Lattices of Dominions in Quasivarieties of Abelian Groups

Let $\mathcal M$ be any quasivariety of Abelian groups, $\operatorname{dom}^{\mathcal M}_G(H)$ be the dominion of a subgroup $H$ of a group $G$ in $\mathcal M$, and $L_q(\mathcal M)$ be the lattice of subquasivarieties of $\mathcal M$. It is proved that $\operatorname{dom}^{\mathcal M}_G(H)$ coincides with a least normal subgroup of the group $G$ containing $H$, the factor group with respect to which is in $\mathcal M$. Conditions are specified subject to which the set $L(G,H,\mathcal M)=\{\operatorname{dom}^{\mathcal N}_G(H)\mid\mathcal N\in L_q(\mathcal M)\}$ forms a lattice under set-theoretic inclusion and the map $\varphi\colon L_q(\mathcal M)\rightarrow L(G,H,\mathcal M)$ such that $\varphi (\mathcal N)=\operatorname{dom}^{\mathcal N}_G(H)$ for any quasivariety $\mathcal N\in L_q(\mathcal M)$ is an antihomomorphism of the lattice $L_q(\mathcal M)$ onto the lattice $L(G,H,\mathcal M)$.