Is there a notation for the repetition of basic operations?

Question

I mean...

• multiplication is the repetition of addition:
  • $2*2 = 2+2$
  • $3*3 = 3+3+3$
• exponential is the repetition of multiplication:
  • $2^2 = 2 * 2$
  • $3^3 = 3*3*3$

.. it is an obvious pattern.

I propose:

• Addition is rank 1
• Multiplication is rank 2
• Exponential is rank 3
• etc ...

This would mean

$H(r, x)$ is the $r$th rank operation on $x$.

For example: $H(3, 2) = 2^2$

and $H(4, 3) = {3^{3^3}}$

It is already kind of wired to get this value since it is quite big. But the function of the rank is certainly one that rises fast.

Does anyone know more about this kind of stuff?

Answer 1

you might check out : https://en.wikipedia.org/wiki/Hyperoperation it talks about different operations to extend this chain of operations.

Answer 2

Note also that you can go even in the opposite direction lowering what you called "rank", in a log-like way. For instance, just like addition is one rank lower than multiplication, you can devise an operation that is one rank lower than addition: the tropical addition, as Vladimir I. Arnold called it.

Answer 3

There is also the notation $a^{n*} $ for $\underbrace {a*\cdots*a}_{\text {$n $ factors} }$. Using this notation we have $$na=a^{n+} $$ $$a^n=a^{n\cdot} $$