Alright, so I finally uncovered the recurrence relation for superfunctions and half-superfunctions!
It can linguistically be put forth as:
the half superfunction of the half superfunction of f is the superfunction of f.
which to me now, is a "no duhhh"; Can't believe I missed it.
This is written using the following notation (I've switched from the diamond notation because it is confusing using it, I'll adopt this transformation notation instead).
\( F^t \{ f \} ( G^t \{ f \} (x)) = x \)
\( F^t \{ f \} (x) = F^{t + 1} \{ f \} (G^{t+1} \{ f \} (x) + 1) \)
\( G^t \{ f \} (x) = F^{t + 1} \{ f \} (G^{t+1} \{ f \} (x) - 1) \)
\( F^0 \{ f \} (x) = f(x) \)
therefore \( F^t \{ f \}(x) \) is the t'th superfunction of \( f \) and \( G^t \{ f \} (x) \) is the t'th abel function of \( f \).
and now we have the recurrence relation:
\( F^m \{ F^n \{ f \} \} (x) = F^{m + n} \{ f \} (x) \)
Giving us what I put forth earlier
the half superfunction of the half superfunction of f is the superfunction of f.
\( F^{\frac{1}{2}} \{ F^{\frac{1}{2}} \{ f \} \} (x) = F \{ f \} (x) \)
This definition cannot fall victim to the method of disproof Tommy gave forth because the transformation notation takes three arguments whereas the diamond notation only takes two.
It can linguistically be put forth as:
the half superfunction of the half superfunction of f is the superfunction of f.
which to me now, is a "no duhhh"; Can't believe I missed it.
This is written using the following notation (I've switched from the diamond notation because it is confusing using it, I'll adopt this transformation notation instead).
\( F^t \{ f \} ( G^t \{ f \} (x)) = x \)
\( F^t \{ f \} (x) = F^{t + 1} \{ f \} (G^{t+1} \{ f \} (x) + 1) \)
\( G^t \{ f \} (x) = F^{t + 1} \{ f \} (G^{t+1} \{ f \} (x) - 1) \)
\( F^0 \{ f \} (x) = f(x) \)
therefore \( F^t \{ f \}(x) \) is the t'th superfunction of \( f \) and \( G^t \{ f \} (x) \) is the t'th abel function of \( f \).
and now we have the recurrence relation:
\( F^m \{ F^n \{ f \} \} (x) = F^{m + n} \{ f \} (x) \)
Giving us what I put forth earlier
the half superfunction of the half superfunction of f is the superfunction of f.
\( F^{\frac{1}{2}} \{ F^{\frac{1}{2}} \{ f \} \} (x) = F \{ f \} (x) \)
This definition cannot fall victim to the method of disproof Tommy gave forth because the transformation notation takes three arguments whereas the diamond notation only takes two.