S map

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 variable of S map, a pair of \(g(m) \in \mathbb{N}\) and a function \(g(x)\) is obtained as return value, 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 defining 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 is the operation of adding \(\omega\) to the ordinal of FGH.