SS map is a function which maps "a set of a natural number and a function and S map" to "a set of a natural number and a function and S map". It was defined by Japanese googologist Fish in 2002[1] and used in the definition of Fish number 1 and Fish number 2.

SS map in \(F_1\)

\begin{eqnarray*} SS:[m,f(x),S]→[n,g(x),S2] \end{eqnarray*} Here, S2, n, and g(x) are defined as follows. \begin{eqnarray*} S2 & = & S^{f(m)} \\ S2 & : & [m,f(x)]→[n,g(x)] \end{eqnarray*}

SS map in \(F_2\)

\begin{eqnarray*} SS:[m,f(x),S]→[n,g(x),S2] \end{eqnarray*} Here, S2, n, and g(x) are defined as follows. \begin{eqnarray*} S2 & = & S^{f(m)} \\ S2:[m,f(x)] & → & [n,p(x)] \\ S2^x:[m,f(x)] & → & [q,g(x)] \\ \end{eqnarray*}

Difference of SS map in \(F_1\) and \(F_2\)

In \(F_1\), SS map repeats S map \(f(m)\) times, while in \(F_2\), SS map diagonizes the repetition time of S map. It looks similar but actually this is a big difference, because the SS map in \(F_1\) has the growing level of \(A(0,1) → A(1,0,1,1) → A(1,0,1,2) → A(1,0,1,3)\), while the SS map in \(F_2\) has the growing level of \(A(0,1)→A(1,0,0,n)→A(2,0,0,n)→A(3,0,0,n)\).

As the essential growing mechanism in \(F_2\) is to "diagonizing the functional", in Fish number 3, the definition was simplified. The idea of SS map, i.e., making higher order functional to produce fast-growing function, was not effective in \(F_1\) and \(F_2\) and therefore discarded in \(F_3\); and it was finally realized effectively in Fish number 5.

Sources

See also

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
By 巨大数大好きbot: Flan numbers
By Jason: Irrational arrow notation · δOCF · δφ · ε function
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
By Yukito: Hyper primitive sequence system · Y sequence · YY sequence · Y function
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's \(\psi\) · Sushi Kokuu Hen

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