So I was sitting extremely bored in class the other day, and I thought about something:
$a \cdot b := \overbrace{a+a+...+a}^{b}$
$a^{b} := \overbrace{a \cdot a \cdot ... \cdot a}^{b}$
$^{b}a := \overbrace{a^{a^{...}}}^{b}$
This can also go the other way:
$a+b := \overbrace{SS...S}^{b}a \, $ (brackets left out for clarity)
However, I could not find any other operators which are defined (or at least started as being defined) as repeating a "lower level operation" (whatever that means).
Is there some sort of function $f:(\mathbb{Z}, \mathbb{Z}, \mathbb{Z}) \rightarrow \mathbb{Z}$ such that: $f(a, b, 0) = \overbrace{SS...S}^{b}a$ $f(a, b, n) = f^{°b}(a, a, n-1)$ where $f^{°n}(p)$ denotes repeated appliance of $f$? If so, how can we generalize it to $f(a, b, n)$ $(n\in\mathbb{Z}^{-})$?
