16.20 (2006)

Let $\mathfrak{M}$ be a quasivariety of groups. The dominion $\text{dom}_A^\mathfrak{M}(H)$ of a subgroup $H$ of a group $A$ (in $\mathfrak{M}$) is the set of all elements $a \in A$ such that for any two homomorphisms $f, g : A \to B \in \mathfrak{M}$, if $f, g$ coincide on $H$, then $f(a) = g(a)$. Suppose that the set $\{\text{dom}_A^\mathfrak{N}(H) \mid \mathfrak{N}$ is a quasivariety, $\mathfrak{N} \subseteq \mathfrak{M}\}$ forms a lattice with respect to set-theoretic inclusion. Can this lattice be modular and non-distributive?