What is this operator called?

Question

If $x \cdot 2 = x + x$

and $x \cdot 3 = x + x + x$

and $x^2 = x \cdot x$

and $x^3 = x \cdot x \cdot x$

Is there an operator $\oplus$ such that:

$x \oplus 2 = x^x$

and $x \oplus 3 = {x^{x^x}}$?

Also, is there a name for such a set of operators ops where...

Ops(1) is addition

Ops(2) is multiplication

Ops(3) is exponentiation

Ops(4) is $\oplus$

...and so on

Also, is there a branch of math who actually deals with such questions? Have these questions already been answered like 2000 years ago?

Answer 1

This operation ${\rm Ops}(4)$ is called tetration, from the greek root tetra meaning four; it's also sometimes called a "power tower". There are also many further generalizations of this type of sequence; Knuth's up-arrow notation gives $a^{a^{a^a}}=a\uparrow\uparrow4$, so that $a\uparrow\uparrow n$ is the tetration operation. By adding more arrows you get pentation and so on, and the Conway chained arrow notation generalizes this still further.

FYI, for "to the power of-ation" the word you're looking for is exponentiation.

Answer 2

A more general function that combines all those operators has been defined by Ackermann:

$ \varphi(m,n,p) = \begin{cases} \varphi(m, n, 0) = m + n \\ \varphi(m, 0, 1) = 0 \\ \varphi(m, 0, 2) = 1 \\ \varphi(m, 0, p) = m &\text{ for } p > 2 \\ \varphi(m, n, p) = \varphi(m, \varphi(m, n-1, p), p - 1) &\text{ for } n > 0 \text{ and } p > 0. \end{cases} $

So for $p = 0, 1, 2$ you get

$\phi(m, n, 0) = m + n $
$\phi(m, n, 1) = m \cdot n $
$\phi(m, n, 2) = m ^ n $

and

$\phi(m, n, 3) = \overbrace{{{m ^ m} ^ m} ^ {...}}^{n}$