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, $

succ:S(x)add:m+nmult:m∗nexp:mnm↑ntet:mnm↑↑nhyper-k:hk(m,n)m↑kn\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

h1(m,n+1)=h0(m,h0(m,n))=S(m+n)h2(m,n+1)=h1(m,h1(m,n))=m+(m∗n)h3(m,n+1)=h2(m,h2(m,n))=m∗(mn)hk′(m,n′)=hk(m,hk′(m,n))=m↑k(m↑k′n′)hk+1(m,n+1)=hk(m,hk+1(m,n))=m↑k(m↑k+1n)hk(m,n)=hk−1(m,hk(m,n−1))=m↑k−1(m↑kn−1)\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:

    H0(m,n)=1+nH_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: 2↑4=2×(2×(2×2))=24=16\displaystyle{ 2 \uparrow 4 = 2 \times (2\times (2\times 2)) = 2^{4} = 16 }

Double arrow is iterated exponentiation i.e. tetration: 2↑↑4=2↑(2↑(2↑2))=42=2222=224=216=65,536\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): 2↑↑↑3=2↑↑(2↑↑2)=2↑↑(2↑2)=2↑↑4=2222\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: m↑kn={1if n=0m↑k−1(m↑k(n−1))otherwise\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} } }

↑k\uparrow^k stands for k arrows: 2↑↑↑↑3=2↑43{\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow^{4}3}

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