M(m,n) map

m(m,n) map is a function which maps maps to maps. It was defined by Japanese googologist Fish in 2007Fish, Googology in Japan - exploring large numbers (2013) and used to define Fish number 6. It is a 2 variable extension of m(n) map and has a growth rate of \(f_{\zeta_0}(n)\).
== Definition ==
\(M[m,n]\) map \((m=0,1,...; n=1,2,...)\) is defined as follows.
* \(M[0,1]\) is a set of functions of natural numbers to natural numbers.
* \(M[m+1,1]\) is a set that has one element from each \(M[m,n] (n=1,2,...)\); \(M[m+1,1]\) is a of \(M[m,1], M[m,2], M[m,3], ...\).
** Element of \(M[m,1]\) also works as a function in \(M[0,1]\) that is element of element of ... element of \(M[m,1]\).
* \(M[m,n+1] (n=1,2,...)\) is a set of maps from \(M[m,n]\) to \(M[m,n]\).
Element of \(M[m,n]\), \(m(m,n)\) is defined as follows. Here, \(a_i, b_i,f_i\) are elements of \(m(m,i)\) and strict structure of definition is same as m(n) map.
\begin{eqnarray*}
m(0,1) (x) & := & x+1 \\
m(m,n+1) f_n f_{n-1} ...f_1 (x) & := & f_n^xf_{n-1}... f_1 (x) \\
& & (m=0; n=1,2,... \text{ or } m=1,2,...; n=2,3,…) \\
m(m+1,1) & := & [m(m,1),m(m,2),m(m,3),…] \\
m(m+1,2)[a_1,a_2,...] & := & [b_1,b_2,…] \text{ where } b_n \text{ is defined as:} \\
b_n f_{n-1}...f_1(x) & := & a_y a_{y-1}...a_n f_{n-1}…f_1(x) (y=max(x,n))
\end{eqnarray*}
== Calculation ==
\begin{eqnarray*}
m(1,1)(x) & = & [m(0,1),m(0,2),m(0,3),…](x) \\
& = & m(0,1)(x) = x+1 \\
\end{eqnarray*}
Let \(m(1,2) m(1,1) = [a_1,a_2,a_3,…]\) and
\begin{eqnarray*}
a_1(x) & = & m(0,x) m(0,x-1) … m(0,1) (x) \\
& \approx & f_{\varepsilon_0}(x) \\
\therefore m(1,2) m(1,1)(x) & \approx & f_{\varepsilon_0}(x) \\
m(0,2) a_1(x) & \approx & f_{\varepsilon_0 + 1}(x) \\
m(0,3) m(0,2) a_1 (x) & \approx & f_{\varepsilon_0 + \omega}(x) \\
m(0,4) m(0,3) m(0,2) a_1 (x) & \approx & f_{\varepsilon_0 + \omega^{\omega}}(x) \\
m(0,5) m(0,4) m(0,3) m(0,2) a_1 (x) & \approx & f_{\varepsilon_0 + \omega^{\omega^{\omega}}}(x) \\
a_2 a_1(x) & = & m(0,y) m(0,y-1) …m(0,2) a_1(x) (y=max(x,2)) \\
& \approx & f_{\varepsilon_0 \times 2}(x) \\
\end{eqnarray*}
Then,
\begin{eqnarray*}
m(0,3) a_2 a_1 (x) & \approx & f_{\varepsilon_0 \times \omega}(x) \\
m(0,4) m(0,3) a_2 a_1 (x) & \approx & f_{\varepsilon_0 \times \omega^{\omega}}(x) \\
m(0,5) m(0,4) m(0,3) a_2 a_1 (x) &\approx & f_{\varepsilon_0 \times \omega^{\omega^{\omega}}}(x) \\
a_3 a_2 a_1(x) & = & m(0,y) m(0,y-1) ... m(0,3) a_2 a_1 (x) (y=max(x,3)) \\
& \approx & f_{\varepsilon_0 ^2}(x)
\end{eqnarray*}
As for \(a_4\),
\begin{eqnarray*}
m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^\omega}(x) \\
m(0,5) m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^{\omega \times 2}}(x) \\
m(0,6) m(0,5) m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^{\omega ^ 2}}(x) \\
m(0,7) m(0,6) m(0,5) m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^{\omega^{\omega}}}(x) \\
a_4 a_3 a_2 a_1(x) & = & m(0,y) m(0,y-1)...m(0,4) a_3 a_2 a_1(x) (y=max(x,4)) \\
& \approx & f_{\varepsilon_0 ^{\varepsilon_0}}(x)
\end{eqnarray*}
And similarly, \(a_5\), \(a_6\) can be calculated as
\begin{eqnarray*}
a_5 a_4 a_3 a_2 a_1(x) & \approx & f_{\varepsilon_0 ^{\varepsilon_0^{\varepsilon_0}}}(x) \\
a_6 a_5 a_4 a_3 a_2 a_1(x) & \approx & f_{\varepsilon_0 ^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(x) \\
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
m(1,2)^2 m(1,1) (x) &=& m(1,2)[a_1,a_2,...](x) \\
& \approx & f_{\varepsilon_1}(x)
\end{eqnarray*}
Now let
\begin{eqnarray*}
m(1,2)^3 m(1,1)(x) = [b_1,b_2,b_3,...](x)
\end{eqnarray*}
and for calculating \(b_i\), \(\varepsilon_0\) in the calculation of \(a_i\) is changed to \(\varepsilon_1\). Therefore,
\begin{eqnarray*}
m(1,2)^3 m(1,1)(x) & \approx & f_{\varepsilon_2}(x) \\
m(1,2)^4 m(1,1)(x) & \approx & f_{\varepsilon_3}(x) \\
m(1,2)^n m(1,1)(x) & \approx & f_{\varepsilon_{n-1}}(x) \\
\end{eqnarray*}
Then calculation goes on similar to \(m(n)\) map.
\begin{eqnarray*}
m(1,3) m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_\omega} \\
m(1,4) m(1,3) m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_{\omega^\omega}} \\
m(1,5) m(1,4) m(1,3) m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_{\omega^{\omega^\omega}}} \\
m(2,2) m(2,1) (x) &=& m(1,x) m(1,x-1) ... m(1,2) m(1,1) (x) \\
& \approx & f_{\varepsilon_{\varepsilon_0}} \\
\end{eqnarray*}
And then
\begin{eqnarray*}
m(2,2)^2 m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_1}}(x) \\
m(2,2)^3 m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_2}}(x) \\
m(2,2)^4 m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_3}}(x) \\
m(2,3) m(2,2) m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_\omega}}(x) \\
m(3,2) m(3,1) (x) &=& m(2,x) m(2,x-1) ... m(2,2) m(2,1) (x) \\
& \approx & f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}(x) 
\end{eqnarray*}
As we have calculated to \(m(3,2) m(3,1) (x)\), it is easy to see that
\begin{eqnarray*}
m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_0} (x) \\
m(2,2) m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_0}} (x) \\
m(3,2) m(3,1) (x) & \approx & f_{\varepsilon_{\varepsilon_{\varepsilon_0}}} (x) \\
m(4,2) m(4,1) (x) & \approx & f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}} (x) \\
\end{eqnarray*}
and therefore
\[m(x,2)m(x,1)(x) \approx f_{\zeta_0}(x)\]
== Sources ==
==See also==
ja:m(m,n)変換
Category:Functions
Category:Googology in Asia
Category:Notations
Category:Numbers by Fish