User blog:Denis Maksudov/Nomenclature for the hyper operators

Hyperoperations can be extended to transfinite ordinals

Let's define

1) $$H(n,1,b) = n+b$$,

2) $$H(n,\alpha+1,b) = \underbrace{H(n,\alpha,H(n,\alpha,H(\cdots H(n,\alpha,n}_{b \quad n's})\cdots)))$$,

3) $$H(n,\alpha,b) = H(n,\alpha [b],n)$$ iff $$\alpha$$ is a limit ordinal.

Or in such form

1) $$n \uparrow^{-1} b = n+b$$,

2) $$n \uparrow^{\alpha+1} b = \underbrace{n \uparrow^{\alpha} (n \uparrow^{\alpha} (\cdots (n \uparrow^{\alpha} n}_{b \quad n's})\cdots))$$,

3) $$n\uparrow^{\alpha} b = n\uparrow^{\alpha [b]} n$$ iff $$\alpha$$ is a limit ordinal.


 * Third rule was inspired by Aeton's work

Example

$$10\uparrow^{\omega} 3 = 10\uparrow^{3} 10$$

$$10\uparrow^{\omega+1} 3 = 10\uparrow^{\omega}(10\uparrow^{\omega}10)=\left. \begin{matrix} &&10 \underbrace{\uparrow\uparrow\uparrow\uparrow\cdots\uparrow\uparrow\uparrow\uparrow}10\\ & &10 \underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}10 \\ & &10 \end{matrix} \right \} \text {3 layers}$$

and so on.

Nomenclature

Abbreviations adopted for the morphological derivation:

add=addition, ult=multiplication, ex=exponentiation, etr=tetration, om=omega, phi - binary Veblen function

un,b,tr,quadr,quint,sext,sept,oct,non,dek=1,2,3,4,5,6,7,8,9,10

(thus for the creation of names of hyperoperations almost same abbreviations were used as in my nomenclature of numbers)

$$a\uparrow^{\varphi(1,0)} b $$ uniphiation,

$$a\uparrow^{\varphi(2,0)} b $$ biphiation,

$$a\uparrow^{\varphi(3,0)} b $$ triphiation,

$$a\uparrow^{\varphi(4,0)} b $$ quadriphiation,

$$a\uparrow^{\varphi(5,0)} b $$ quintiphiation,

$$a\uparrow^{\varphi(6,0)} b $$ sextiphiation,

$$a\uparrow^{\varphi(7,0)} b $$ septiphiation,

$$a\uparrow^{\varphi(8,0)} b $$ octiphiation,

$$a\uparrow^{\varphi(9,0)} b $$ noniphiation,

$$a\uparrow^{\varphi(10,0)} b $$ dekophiation

and so on.