Definition:Hyperoperation
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Definition
Hyperoperation Sequence
The hyperoperation sequence is the sequence $\left\langle{H_n}\right\rangle$ of binary operations $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, defined as:
$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
$n$th Hyperoperation
The $n$th term of $\left\langle{H_n}\right\rangle$, which is the binary operation $H_n : \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{\ge 0}$, is known as the $n$th hyperoperation.
Also known as
Some sources refer to this as the hyperoperator.