A new set of numbers is necessary to extend tetration to real exponents.

[marraco]

....
Tetration exponent product is non commutative. In general:

\( ^n(^{\frac{1}{m}}a) \,\neq\, ^{\frac{1}{m}}(^na) \)

I would also add that for analytic tetration in general:
\( ^n(^{\frac{1}{n}}a) \,\neq a \)

I defined \( ^n(^{\frac{1}{n}}a) \, = a \)

as ttr(\( (^{\frac{1}{n}}a) \) ,n,1)=a

were \( (^{\frac{1}{n}}a) \) may be multivalued.

It has some problems, like \( \lim_{n \rightarrow \infty} (^{\frac{1}{n}}a) = a^{\frac {1}{a}} \), which has "a" solutions, (for a E N), against °a=1 by definition.

So, it looks like this definition of \( ^{\frac{1}{m}}a \) is wrong all the way.

Still, there should be some, non integer number t, such

\( c \,=\, ({^t{a}}) \,\Leftrightarrow\, ^nc\,=\, a \)