Dominions of universal algebras and projective properties

Let $A$ be a universal algebra and $H$ its subalgebra. The dominion of $H$ in $A$ (in a class $\mathcal M$) is the set of all elements $a\in A$ such that every pair of homomorphisms $f,g\colon A\to M\in\mathcal M$ satisfies the following: if $f$ and $g$ coincide on $H$, then $f(a)=g(a)$. A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras $H$ whose dominions coincide with $H$. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup $H$ is closed in each group $\langle H,a\rangle$ generated by one element modulo $H$.