Notations

There are an unnecessarily large number of notations in common use for functions related to tetration. Ideally, only the most powerful notations would be used, however, as a matter of convinience we use left-superscript notation in preference to arrow notation and tower notation. Also, even though box notation is closer to hyper-operation terminology, arrow notation will be used, because it is more well-known.

Arrow notation is used to write hyper-operations, and looks like $x{\U}{\U}n$ for tetration. For more information, see the Hyperops page.
Iteration notation is used to write iterated functions, and looks like $\iter{f}{n}(x)$
Quasigroup notation is used to write inverse functions of binary operations, and it looks like $(/*)$ (for hyper-roots) and $(*{\backslash{}})$ (for hyper-logs).
Section notation is used to write binary operations applied to only one argument, and it looks like $(x*)$ or $(*y)$ . For more information, see Haskell sections.
Tower notation is used to write nested exponentials, and looks like $\bigT_{k=1}^{n}a_k$

When all exponents are equal, and the top is 1, this can be written with left-superscript notation for tetration.
When all exponents are equal except for the top, then this can be written with iterated exponential notation.

When choosing a notation, keep in mind your audience, and familiarity with tetration. In general, tetration should be written with left-superscript notation. If there is confusion with multiples in front, then use tower notation $\bigT^{n}a$ , or if there is a complicated expression in the height, then use arrow notation $x{\U}{\U}n$ .

Axioms

$a \U\U n = a \U {\left( a \U\U (n-1) \right)}$

$(a{\U\U})^{-1}(a^z) = (a{\U\U})^{-1}(z) + 1$

Infinite Tetrate and Company

The infinite tetrate is topologically conjugate to the 2nd super-root and the Lambert W-function.

${}^{\infty}x & = \frac{W(-\ln(x))}{-\ln(x)} = \exp(-W(-\ln(x))) = \frac{1}{\ssqrt{1/x}} \\\notag \frac{1}{{}^{\infty}(1/x)} & = \frac{\ln(x)}{W(\ln(x))} = \exp(W(\ln(x))) = \ssqrt{x} \\\notag {}^{\infty}(\ee^{-x})x & = W(x) = x\exp(-W(x)) = \frac{x}{\ssqrt{\ee^x}} \\\notag$

for more information see the section on topological conjugacy.

Table of Tetration

For integer values:

${}^{-2}a & = \frac{-\infty}{\ln(a)} \\\notag {}^{-1}a & = 0 \\\notag {}^{0}a & = 1 \\\notag {}^{1}a & = a \\\notag {}^{2}a & = a^a \\\notag {}^{3}a & = a^{a^a} \\\notag$

For infinite values:

${}^{{-}\infty}x & = \frac{W_{-1}(-\ln(x))}{-\ln(x)} \quad\text{for all } x>\ee^{1/\ee}\\\notag {}^{{-}\infty}x & = \frac{W(-\ln(x))}{-\ln(x)} \quad\quad\text{for all } x<\ee^{-\ee} \\\notag {}^{\infty}x & = \frac{W(-\ln(x))}{-\ln(x)} \quad\quad\text{for all } \ee^{-\ee} \le x \le \ee^{1/\ee}$

Table of Iterated Exponentials

For integer values from any x:

$\exp_a^{-2}(z) & = \log_a(\log_a(z)) \\\notag \exp_a^{-1}(z) & = \log_a(z) \\\notag \exp_a^{0}(z) & = z \\\notag \exp_a^{1}(z) & = a^z \\\notag \exp_a^{2}(z) & = a^{a^z} \\\notag \exp_a^{3}(z) & = a^{a^{a^z}} \\\notag$

For integer values from -1:

$\exp_a^{-2}({-}1) & = \log_a(\log_a(-1)) \\\notag \exp_a^{-1}({-}1) & = \frac{\ii \pi}{\ln(a)} \\\notag \exp_a^{0}({-}1) & = -1 \\\notag \exp_a^{1}({-}1) & = \frac{1}{a} \\\notag \exp_a^{2}({-}1) & = a^{1/a} \\\notag \exp_a^{3}({-}1) & = a^{a^{1/a}} \\\notag$