Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of $X$, we have that if $X_0,\ldots,X_{n-1}$ are substructures of $X$, then so too is $f(X_0,\ldots,X_{n-1});$ because after all homomorphisms preserve substructures.
Question. Is there a name for the algebraic structure induced by the operations of $X$ on the set of all substructures of $X$? A link or reference to some more information would be really nice.