SS map

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 making higher order functional was discarded in \(F_3) and finally realized effectively in \(F_5\).