# Hyperoperations

* Level 0
  * Successor
  * Predecessor
* Level 1
  * Addition
  * Subtraction
  * overflow add/sub
  * bound add/sub
  * Summation
* Level 2
  * Multiplication
  * Division
  * Product
* Level 3
  * Exponentiation
  * nth root
  * Logarithm
* Level 4
  * Tetration
  * Super-root
  * Super-logarithm
* Level k
  * hyper-k hyperoperations
  * pentation
  * hexation
  * -ation
  * hepta, octa, nava, deka

factorial super-factorial, $

$$
\begin{align}
& \text{succ}:    & S(x)     \quad &                       \\
& \text{add}:     & m+n      \quad &                       \\
& \text{mult}:    & m\*n      \quad &                       \\
& \text{exp}:     & m^n      \quad & m \uparrow         n  \\
& \text{tet}:     & ^mn      \quad & m \uparrow\uparrow n  \\
& \text{hyper-k}: & h\_k(m,n) \quad & m \uparrow^{k}   n    \\
\end{align}
$$

### Hyperops

$$
\begin{align}
& h\_{1}  (m,n\_{+1})&=& h\_{0}(m,h\_0    (m,n)) &=& S(m+n)        \tag{add} \\
& h\_{2}  (m,n\_{+1})&=& h\_{1}(m,h\_1    (m,n)) &=& m+(m*n)       \tag{mul} \\
& h\_{3}  (m,n\_{+1})&=& h\_{2}(m,h\_2    (m,n)) &=& m*(m^n)       \tag{exp} \\
& h\_{k'}(m,n')&=& h\_k(m,h\_{k'}(m,n)) &=& m\uparrow^{k}(m\uparrow^{k'}n') \\
& h\_{k+1}(m,n\_{+1})&=& h\_{k}(m,h\_{k+1}(m,n)) &=& m\uparrow^{k}(m\uparrow^{k+1}n) \\
& h\_{k}(m,n)&=& h\_{k-1}(m,h\_k(m,n\_{-1})) &=& m\uparrow^{k-1}(m\uparrow^{k}n-1)\\
\end{align}
$$

## Zeration, H0

* 0th hyperop
* successor function, successor operator
* succession

  As the zeroth hyperoperation, successor is also called **zeration**: 

  $$H\_0(m, n) = 1 + n$$

The successor function is the level-0 foundation of the infinite Grzegorczyk hierarchy of hyperoperations, used to build addition, multiplication, exponentiation, tetration, etc. The extension of zeration is addition, defined as *repeated succession*. The extension of addition is multiplication, defined as *repeated addition*.

### Knuth's up-arrow notation

Single arrow is iterated multiplication i.e. exponentiation:\
$$\displaystyle{ 2 \uparrow 4 = 2 \times (2\times (2\times 2)) = 2^{4} = 16 }$$

Double arrow is iterated exponentiation i.e. tetration:\
$$\displaystyle{ 2\uparrow \uparrow 4 = 2 \uparrow (2\uparrow (2\uparrow 2)) = {^4}2 = 2^{2^{2^{2}}} = 2^{2^{4}} = 2^{16} = 65,536 }$$

Triple arrow is iterated tetration (pentation):\
$$\displaystyle{ \begin{aligned} 2\uparrow \uparrow \uparrow 3 &=2\uparrow \uparrow (2\uparrow \uparrow 2) \ &=2\uparrow \uparrow (2\uparrow 2) \ &=2\uparrow \uparrow 4 \ &= {^{^{^{^2}2}2}2} \end{aligned} }$$

The general definition of the notation:\
$$\displaystyle{ m \uparrow^k n = { \begin{cases} 1 & \text{if } n=0 \ m \uparrow^{k-1}(m \uparrow^{k}(n\_{-1})) &{\text{otherwise}} \end{cases} } }$$

$$\uparrow^k$$ stands for k arrows:\
$${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow^{4}3}$$