I was pondering, could one define a function such that
$$T(a,b)^{T(c,d)}=T(a^c,b^d)$$
$∀a,b,c,d$?
I asked the same question some minutes ago, and someone was able to find an example, that of $T(x,y)=1, ∀x,y$. And as much as this is completely correct, I was wondering if more not-so-trivial examples exist.
I'll add a couple of questions to the post then.
- Could one define a function as the latter which maps $T(a,b)$ to a number $w$ of our choosing?
For instance, would it be possible to have $T(8,25) = e$? Notice that by defining $T(8,25)$, one is also in turn defining $T(8^8,25^{25})$, among other infinitely many other pairs of values on $T$. I'm not sure however, if by defining $T(8, 25)$ one is also defining $T(2,2)$, for example, which leads to my next question.
- Assuming that the answer to question $1$ is "yes", is it then possible to map, again, another pair of elements in $T$ to a number of our choosing, say...$T(2,2)=π$? Or would this lead to a contradiction?
$T(1,1)$ would definitely have to be equal to $1$, since $$T(a,b)^{T(1,1)}=T(a^1,b^1)$$
I would truly appreciate any help/thoughts!