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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.

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+1 Good question. I don't know a name for such things. Side question: when you say "substructures of X" do you simply mean "subalgebras of X," or do you mean something more general, maybe encompassing relational structures as well? –  William DeMeo Feb 4 at 1:49

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