If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor $F_X : {\cal A} \rightarrow {\cal B}$, that maps $A \in \cal A$, to $A / \sim $, where $\sim$ is the congruence on $A$ generated by $X$ (so for groups and abelian groups this functor is abelianization). Moreover, any homomorphism from $A \in \cal A$ to $B \in \cal B$ factors uniquely through $F_X(A)$. Is there a standard name for this construction?
I am looking at a situation where there is a natural embedding of $F_X(A)$ in $A$, i.e., the natural $\cal A$-homomorphism from $A$ to $F_X(A)$ factors as the composite of an endomorphism of $A$ and an isomorphism. My example generalises the case where $\cal A$ is the variety of brouwerian semilattices, $\cal B$ is the subvariety defined by the identity $\lnot\lnot x \simeq x$ and the endomorphism is $x \mapsto \lnot\lnot x$. Has this kind of situation been studied in general, or does anyone know of any other specific examples? Any references are greatly appreciated.
