Distributivity Conditions for Lattices of Dominions in Quasivarieties of Abelian Groups

Let $\mathcal{M}$ be any quasivariety of Abelian groups, $L_{q}(\mathcal{M})$ be a subquasivariety lattice of $\mathcal{M}$, ${\rm dom}^{\mathcal{M}}_{G}(H)$ be the dominion of a subgroup $H$ of a group $G$ in $\mathcal{M}$, and $G/{\rm dom}^{\mathcal{M}}_{G}(H)$ be a finitely generated group. It is known that the set $L(G,H,\mathcal{M})=\{{\rm dom}^{\mathcal{N}}_{G}(H)\mid \mathcal{N}\in L_{q}(\mathcal{M})\}$ forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of ${\rm dom}^{\mathcal{M}}_{G}(H)$. It is proved that the lattice $L(G,H,\mathcal{M})$ is semidistributive and necessary and sufficient conditions are specified for its being distributive.