S(n) map

s(n) map is a , which is a function which maps functions to functions. It was defined by Japanese googologist Fish in 2002Fish, Googology in Japan - exploring large numbers (2013) and used to define Fish number 3. The name of the map was taken from the Japanese word shazou, which means mapping.
==Definition==
\begin{eqnarray*}
s(1)f & := & g; g(x)=f^x(x) \\
s(n)f & := & g; g(x)=[s(n-1)^x]f(x) (\text{if }n>1)
\end{eqnarray*}
==Analysis==
Let \(f(x) = x+1\), and the growth rate can be calculated as:
\begin{eqnarray*}
f^2(x) & = & x+2 \\
f^3(x) & = & x+3 \\
s(1)f(x) & = & f^x(x) = x+x = 2x \\
s(1)^2f(x) & = & g^x(x) = 2^x x > 2^x\text{, where }g(x)=2x \\
s(1)^3f(x) & > & h^{x}(x) > 2 \uparrow^2x \text{, where } h(x)=2^x \\
s(2)f(x) & = & s(1)^{x}f(x) > 2\uparrow^xx>A(x,x)=A(1,0,x) \approx f_{\omega}(x)
\end{eqnarray*}
Similar calculation can be written with the method of the asFish "巨大数の世界 (The world of googology)" 数学セミナー (Mathematics seminar) July, 2019. pp. 28--31.
\begin{eqnarray*}
f & = & \lambda x.x+1 \\
f^2 & = & ff = (\lambda x.x+1)f = f+1 = \lambda x.x+2 \\
f^n & = & \lambda x.x+n \\
s(1)f & = & \lambda x.2x \\
s(1)^2f & = & s(1)(\lambda x.2x) = \lambda x.2^x x \approx \lambda x.2^x \\
s(1)^3f & \approx & s(1)(\lambda x.2^x) \approx \lambda x.2 \uparrow ^2 x \\
s(1)^4f & \approx & s(1)(\lambda x.2\uparrow ^2 x) \approx \lambda x.2 \uparrow ^3 x \\
s(1)^nf & \approx & s(1)(\lambda x.2\uparrow ^{n-1} x) \approx \lambda x.A(n+1,x)
\end{eqnarray*}
Here, \(A\) is Taro's multivariable Ackermann function, where the growth rate in FGH is:
\begin{eqnarray*}
A(..., a3, a2, a1, a0, n) \approx f_{... + \omega^3･a3 + \omega^2･a2 + \omega･a1 + a0}(n)
\end{eqnarray*}
Let \(f(n) = A(X, b, n)\) (X is a vector in any length), and:
\begin{eqnarray*}
A(X, b+1, n) & = & A(X, b, A(X, b+1, n-1)) \\
& = & f(A(X , b+1, n-1)) \\
& = & f^2(A(X, b+1, n-2)) \\
& = & … = f^n(A(X, b+1, 0)) \\
& \approx & f^n(n)
\end{eqnarray*}
Therefore, comparing the 3 functions,
* \(s(1)f(x) = f^x(x)\)
* \(f_{\alpha+1}(n) = f^n_\alpha(n)\)
* \(A(X, b+1, n) = f^n(n)\) where \(f(n) = A(X, b, n)\)
they all have similar growth rate. \(s(1)\) map has the same effect of adding 1 to the ordinal in FGH and adding 1 to the second parameter from right in the Ackermann function. This results in:
\begin{eqnarray*}
s(1)s(2)f(x) & \approx & A(1,1,x) \approx f_{\omega + 1}(x) \\
s(1)^2 s(2)f(x) & \approx & A(1,2,x) \approx f_{\omega + 2}(x) \\
s(1)^n s(2)f(x) & \approx & A(1,n,x) \approx f_{\omega + n}(x)
\end{eqnarray*}
and by diagonizing \(s(1)\) again,
\begin{eqnarray*}
s(2)^2 f(x) = s(1)^x s(2)f(x) \approx A(1,x,x) = A(2,0,x) \approx f_{\omega \times 2}(x)
\end{eqnarray*}
Here, the calculation of \(s(2)^2f(3)\) goes as follows:
\begin{eqnarray*}
s(2)^2f(3) &=& s(1)^3s(2) f(3) \\
&=& [s(1)^2 s(2)f]^3(3) \\
&=& [s(1)^2 s(2)f]^2[[s(1)s(2)f]^3(3)]
\end{eqnarray*}
For this calculation, by changing \(s(2)^2 f\) to \(f_{\omega \times 2}\), \(s(1)^3 s(2)f\) to \(f_{\omega+3}\), \(s(1)^2s(2)f\) to \(f_{\omega+2}\), and \(s(1)s(2)f\) to \(f_{\omega+1}\), respectively, the following is obtained:
\begin{eqnarray*}
f_{\omega \times 2}(3) &=& f_{\omega+3}(3) \\
&=& f_{\omega+2}^3(3) \\
&=& f_{\omega+2}^2(f_{\omega+1}^3(3))
\end{eqnarray*}
which shows exactly how FGH is calculated.
Calculation goes in the same way:
\begin{eqnarray*}
s(2)^n f(x) & \approx & A(n,0,x) \approx f_{\omega \times n}(x) \\
s(3)f(x) & = & s(2)^{x}f(x) \approx A(x,0,x) = A(1,0,0,x) \approx f_{\omega^2}(x) \\
s(3)^2 f(x) & \approx & A(2,0,0,x) \approx f_{\omega^2 \times 2}(x) \\
s(3)^n f(x) & \approx & A(n,0,0,x) \approx f_{\omega^2 \times n}(x) \\
s(4)f(x) & = & s(3)^{x}f(x) \approx A(x,0,0,x) = A(1,0,0,0,x) \approx f_{\omega^3}(x) \\
s(1)^4 s(2)^3 s(3)^2s(4)f(x) & \approx & A(1,2,3,4,x) \approx f_{\omega^3+\omega^2 \times 2+\omega \times 3 + 4}(x) \\
s(5)f(x) & \approx & f_{\omega^4}(x) \\
s(6)f(x) & \approx & f_{\omega^5}(x) \\
s(n)f(x) & \approx & f_{\omega^{n-1}}(x) \\
s(x)f(x) & \approx & f_{\omega^\omega}(x)
\end{eqnarray*}
Therefore, by applying \(s(x)\) map, which diagonizes \(s(n)\) map, to the function \(f(x)=x+1\), the growth rate is \(f_{\omega^\omega}(x)\), similar to array notation and Taro's multivariable Ackermann function.
== Sources ==
==See also==
ja:s(n)変換
Category:Functions
Category:Googology in Asia
Category:Notations
Category:Numbers by Fish