On interpretations of normal modal logics and additive formulas in S4, S5, Grz, K and GL

Let $L$ be a modal logic, $A(p)$ be a formula with a single variable $p$. Consider the translation of formulas $f_A$ that commutes with Boolean connectives and maps a formula of the form $\Diamond B$ to $A(f_A(B))$. Let $L_A$ denote the set of formulas whose translation we derive in $L$. It is easy to verify that $L_A$ is a logic (generally speaking, not a normal one). We will say that the formula $A$ defines the interpretation of $L_A$ in $L$.
It is easy to see that the logic $L_A$ is normal if and only if
$$L \vdash A(p \lor q) \leftrightarrow A(p) \lor A(q) \text{ and } L \vdash A(\bot)\leftrightarrow \bot.$$
We will call such formulas $A$ additive. It turns out that there are only few nonequivalent additive formulas. There are only three in S5, five in S4 and Grz. There are infinitely many additive formulas in K and GL, but they are all relatively simple.
We will discuss the methods by which the mentioned results were obtained and their possible generalizations.