Holomorphic semi operators, using the beta method

[MphLee]

Now, we can continue this operation:
\[
x\,[s]_n\,y = x\,[s+1]_{n-1}\,\left( (x\,[s+1]_{n-1}^{-1}\,y) +1\right)\\
\]

Up to this point everything seems right. using the inversion in the monoid of binary operations under left bracketing your formula translates as \([s]_{n+1}=[s+1]_nS[s+1]_n^{-1}\)

The really weird part now, is that this solution doesn't work on its own. You have to solve for \([s+1]_n\) while you solve for \([s]_n\). The thing is... we can solve for:
\[
\begin{align}
x\,[s]_{n}\,y &= f(y)\\
x\,[s+1]_n\,y &= f^{\circ y}(q)\\
\end{align}
\]

You can actually do this pretty fucking fast... It just looks like iterating a linear function.

EDIT:ACk! made a small mistake here, this is an idempotent iteration like this, the actual iteration is a little more difficult, I'll write it up when I can make sense of controlling the convergence of this...

This last formula look suspicious. Maybe it is related to your small mistake.

We should have something like  \(x\,[s]_{n+1}\, y=f(y)\) and \(x\,[s+1]_{n}\, y=f^y(q)\)