Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "word" with "term" appropriately; I've phrased this question for groups since I think it's most likely to be known in this setting, but I'm really interested in the universal algebraic version.
I've started thinking about a particular "equation-preserving" quotient of the free product of groups (or more generally, coproduct of appropriate algebraic structures). Let $\mathcal{G},\mathcal{H}$ be groups with disjoint underlying sets $G,H$ respectively. Say that a pair of $G\sqcup H$-words $w(g_1,...,g_m,h_1,...,h_n)$ and $v(g_1,...,g_m,h_1,...,h_n)$ are linked iff we have both $$\mathcal{G}\models \forall x_1,...,x_n[w(g_1,...,g_m, x_1,...,x_n)=v(g_1,...,g_m, x_1,...,x_n)]$$ and $$\mathcal{H}\models\forall x_1,...,x_m[w(x_1,...,x_m, h_1,...,h_n)=v(x_1,...,x_m,h_1,...,h_n).$$
For example, if $g\in Z(\mathcal{G})$ and $h\in Z(\mathcal{H})$ then $gh$ and $hg$ are linked. Linkage gives rise to a congruence $\sim$ on the free product $\mathcal{G}*\mathcal{H}$, namely the transitive closure of the relation $\sim_0$ given by $x\sim_0y$ iff there are linked words $w,v$ which evaluate to $x,y$ respectively in $\mathcal{G}*\mathcal{H}$. Let $\mathcal{G}\star \mathcal{H}=\mathcal{G}*\mathcal{H}/\sim$.
Intuitively, the group $\mathcal{G}\star \mathcal{H}$ connects similar "regions" of equational behavior in $\mathcal{G}$ and $\mathcal{H}$. For example, there is a canonical embedding of $Z(\mathcal{G})\times Z(\mathcal{H})$ into $\mathcal{G}\star\mathcal{H}$, but this isn't true for the free product or even the free product "modded out by" the common equational theory of $\mathcal{G}$ and $\mathcal{H}$.
Question: Does this construction have a name?
