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

Some updates:

Suppose there exists as $u \in \Bbb{C}$ such that $H(x,u) = x$ 

Then the solution to the system:

$$ \begin{pmatrix} \theta ( H(x,w), y) = H(\theta(x,y), K_1 ) \\ \theta ( x, H(y,z)) = H(\theta(x,y), K_2) \end{pmatrix}$$ 

for $\theta (x,y) $ with suitable choice of $K_1$ and $K_2$ yields the $\theta(x,y)$ that satisfies 

$$ \begin{pmatrix} \theta ( H(x,w), y) = H(\theta(x,y), \theta(w,u ) \\ \theta ( x, H(y,z)) = H(\theta(x,y), \theta(u,z) ) \end{pmatrix}$$ 

Which must be a solution to: 

$$ \theta(H(x,w), H(y,z)) = H(\theta(x,y), \theta(w,z)) $$ 

So all that remains to be considered is if no such $u$ exists for a given $H$