What properties carry over when making pointwise definitions? Let $A$ be a set with some structure namely (possibly infinitary) operations $(m_k : A^{I_k} \to A)_{k=1}^m$ and relations $(R_l : A^{a_l} \to A)_{l=1}^n$ (let's only discuss finitary ones).
Let $X$ be any set and consider the set of functions $A^X$. Define:
$$\tilde{m}_k : (A^X)^{I_k} \to A^X, (f_i)_{i\in I_k} \mapsto  (x\mapsto m_k(f_ix)_{i\in I_k})$$
and 
$$\tilde{R}_l = \{(f_1,\dots, f_{a_l}) : \forall x\in X :(f_1x,\dots,f_{a_l}x)\in R_l \}$$
Consider the language $\mathcal{L}$ the first-order language associated with $A$ its operations and relations and the language $\mathcal{L}'$ of $A^X$ together with the operations ($\tilde{m}_k)_{k=1}^m$ and relations $(\tilde{R}_l)_{l=1}^n$. 
Given a proposition $P\in \mathcal{L}$ define $\tilde{P}\in \mathcal{L}'$ by replacing all occurences of $m_k$ by $\tilde{m}_k$ and $R_l$ by $\tilde{R}_l$.

What is the greatest subset $L$ of $\mathcal{L}$ such that for all $P\in L$ we have: $$P \text{ holds} \Rightarrow \tilde{P} \text{ holds}$$


I hope I got most things write while writing this but please tell me if something is completely off.
 A: What you call a set with some structure is generally referred to as a structure in first-order logic and model theory. If $A_x$ (where $x$ ranges over some non-empty index set $X$) is any family of structures for a language $\cal L$, you can form a structure for $\cal L$ whose elements are functions $f : X \to \bigsqcup_{x\in X} A_x$ (where $\bigsqcup$ denotes the disjoint union) such that $f(x) \in A_x$, for all $x \in X$, with the operations and relations defined pointwise. This structure is called the direct product of the $A_x$ (or just product for short). You are looking at the special case when the $A_x$ are all the same, namely your original structure $A$. (You might call this special case a direct power of $A$.) Products have been studied widely in universal algebra and model theory. E.g., see Chapter 9 of Wilfrid Hodges' Model Theory or Chapter 7 of Burris and Sankappanavar's A Course in Universal Algebra.
I don't think there can be a useful answer to your question in general. Horn sentences are an important class of sentences that are preserved in products (see Hodges or Burris and Sankappanavar), but for a specific structure many more sentences may be preserved. As a trivial example, a power of a structure $A$ with just one element is isomorphic to $A$ and so all sentences are preserved. As a more interesting example, if the $A_x$ are rings of characteristic $p$, then so is their product, and this could lead to many non-Horn formulas being preserved.
A couple of detailed points: your relations $R_l$ should be subsets of $A^{a_l}$ or functions $A^{a_l} \to 2$ where $2$ is your favourite set with two elements. Rather than change the language, it would be more usual (and technically easier) to talk about different interpretations of the same language in two different structures (in your case, the original structure $A$ and the power structure $A^X$).
