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

\begin{eqnarray*} S:[m,f(x)]→[g(m),g(x)] \end{eqnarray*}

which means that when a pair of $$m \in \mathbb{N}$$ and a function $$f(x)$$ is given as input variables of S map, a pair of $$g(m) \in \mathbb{N}$$ and a function $$g(x)$$ is obtained as return values, where $$g(x)$$ is defined as

\begin{eqnarray*} B(0,n) & = & f(n) \\ B(m+1,0) & = & B(m, 1) \\ B(m+1,n+1) & = & B(m, B(m+1, n)) \\ g(x) & = & B(x,x) \end{eqnarray*}

and $$g(m)$$ is calculated by substituding $$x=m$$ to $$g(m)$$.

$$B(m,n)$$ is similar to Ackermann function except $$B(0,n) = f(n)$$.

## Approximation in other notation

S map is similar to Taro's multivariable Ackermann function with 3 variables. By applying S map n times to [3,x+1], we get a number $$A(n,1,1)$$ and a function $$A(n-1,x,x)$$. Therefore, S map corresponds to adding $$\omega$$ to the ordinal of FGH. At the time when $$F_1$$ was developed, people at Japanese BBS didn't know FGH or multivariable Ackermann function (which was developed in 2007), but it was soon calculated that applying S map is similar to adding one to the length of the arrow of Chained arrow notation.

S map is used in $$F_1$$ and $$F_2$$, but not in Fish number 3, where s(n) map is used instead. $$F_1$$ and $$F_2$$ is based on S map, but later Fish found that s(2) map, which is similar to S map, is obtained with the definition of s(n) map, and the Ackermann function is not necessary in the definition.

## 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