Zeration

[marraco]

f and g are some function :

f^[a - N_q](N_q[q]b) = g^[b - N_q](N_q[q]a)

General case \( f^{a - N_q}(N_q[q]b) = g^{b - N_q}(N_q[q]a) \)

maybe this one is clearer?
\( f^{a}(N_{(q,a)}) = g^{b}(N_{(q,b)}) \)
\( f(x)=x {\small{[q-1]}} b \)
\( g(x)=x {\small{[q-1]}} a \)

I added a subindex to the neutral, because many operations have a different neutral for each number.

For example, right bracket tetration:
\( (..((a_1)^{a_2})^{a_3}^ \,...)^{a_b}=a^{a^{b-1}} \), where
\( f_{(x)}=x {\small{[3]}} b = x^b \)
\( g_{(x)}=x {\small{[3]}} a = x^a \)

...looks non intuitive. Unnecessarily complicated.
But each "a" has a neutral \( N_{()4,a}\,=\, a^{a^{-1}} \), which simplifies the notation:

\( (..((({N_a)}^{a_1})^{a_2})^{a_3}^ \,...)^{a_b}={N_a\,}^{a^{a^b}} \)

f^[a - oo](...) does not make sense as a nonconstant function.

a+b = b+a

"add+1"^[a](0) = "add+1"^[b ](0)