**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^{[1]} and used in the definition of Fish number 1 and Fish number 2. It is defined as

\begin{eqnarray*} S:[m,f(x)]â[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 Edit

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.^{[2]}

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 Edit

- â Fish, Googology in Japan - exploring large numbers (2013)
- â Log of Japanese BBS, October 5, 2002

## See also Edit

**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:** Pair sequence number · Bashicu matrix system**Indian counting system:** Lakh · Crore · Uppala**Chinese and Japanese counting system:** Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu**Buddhist text:** Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra**Other:** Taro's multivariable Ackermann function · Sushi Kokuuhen