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 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

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's $$\psi$$
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 computation programmes · TR function (I0 function)
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Bashicu matrix system 2.3
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 Naruyoko Naruyo: Y sequence computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence · YY sequence · Y function · ω-Y sequence