Given an algebraic structure $A$ (call its underlying set $U$) we can obtain a new algebraic structure $B$ with underlying set $V=\mathcal{P}(U)$ in the obvious way. In particular, if $f : U^n \rightarrow U$ is an operation of $A$, then we have a corresponding operation $f' : V^n \rightarrow V$ defined as follows.
$$f'(X_0,\cdots,X_{n-1}) = \{f(x_0,\cdots,x_{n-1}) : x_0 \in X_0, \cdots x_{n-1} \in X_{n-1}\}.$$
I think that in the case where $f$ is a nullary function (i.e. a constant), the above definition reduces to the following.
$$f' = \{f\}.$$
Anyway, since the signature of $A$ is identical to that of $B$, we can ask:
Question. Which identities are preserved in the passage from $A$ to $B$?
Discussion. It is well known that the associativity and commutativity of any binary operations in the signature will necessarily survive the journey. So I'm thinking that, if every variable occurs precisely once on each side of an identity, then the identity should be preserved. I do not know whether this condition is necessary.