So I was considering the following functional equation:

Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that

$$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), \theta(b,d)) $$

This resulted as a bit of a generalization on the solution to, given $t(x)$ find $H(x,y)$ such that

$$ t(H(x,y)) = H(t(x),t(y))$$

Which has an elegant series solutions using the theory of finite differences. By observing we can split up to any concrete case where:

$$ \begin{pmatrix} H(t(x),y) = q_1(H(x,y)) \\ H(x,t(y)) = q_2(H(x,y)) \\\text{such that} \ q_1(q_2) = t\end{pmatrix} $$

And then recover the 2-d plane of terms that when summed together form $H$.

Does anyone know if others have worked on the same equation? Also what subject is this? I've mostly been working in my black hole, and operator theory+ functional equations seem relevant but i'm not too sure if this is of interest to others