M(n) map

m(n) map is a function which maps maps to maps. It was defined by Japanese googologist Fish in 2003 and used to define Fish number 5. It has a growth rate of \(f_{\varepsilon_0}(n)\), similar to Hydra function and tetrational array of BEAF. The name of the map was taken from a word mapping.

\(m(1)\) is a function which maps from numbers to numbers, and \(m(2)\) is a function which maps from functions to functions, defined as

\begin{eqnarray*} m(1)(x) & = & x^x \\ m(2)f(x) & = & f^x(x) \end{eqnarray*}

where \(m(2)\) matches \(s(1)\) of s(n) map; \(m(2)=s(1)\).

\(m(3)\) map is a function which maps "function which maps functions to function" to "function which maps functions to function", defined and calculated as (parenthesis is written verbosely here to indicate the order of calculation)

\begin{eqnarray*} ((m(3)m(2))f)(x) & = & (m(2)^xf)(x) \\ & = & (s(1)^xf)(x) \\ & = & (s(2)f)(x) \end{eqnarray*} Therefore, \(m(3) m(2) = s(2)\) in \(s(x)\) map, and \(m(3)^2 m(2)\) is calculated as \(s(3)\) as follows. \begin{eqnarray*} ((m(3)^2m(2))f)(x) & = & ((m(3)(m(3)m(2)))f)(x) \\ & = & ((m(3)m(2))^xf)(x) \\ & = & (s(2)^xf)(x) \\ & = & (s(3)f)(x) \end{eqnarray*}

Calculation goes on similarly, with FGH approximation,

\begin{eqnarray*} m(3)^3 m(2) m(1)(x) & = & s(4)f(x) \approx f_{\omega^3}(x) \\ m(3)^4 m(2) m(1)(x) & = & s(5)f(x) \approx f_{\omega^4}(x) \\ m(3)^n m(2) m(1)(x) & = & s(n+1)f(x) \approx f_{\omega^n}(x) \\ m(3)^x m(2) m(1)(x) & = & s(x)f(x) \approx f_{\omega^\omega}(x) \\ \end{eqnarray*}

where \(f(x) = m(1)(x) = x^x\). In general


 * \(M_0\) = set of positive integers
 * \(M_{n+1}\) = set of functions which map \(M_n\) to \(M_n\)

\(m(n)\) map \((n \ge 1)\) belongs to \(M_n\) and is defined as follows.
 * For \(f_n \in M(n)\), \(m(n+1)(f_n) = g_n\) is defined as follows.
 * For \(f_{n-1} \in M(n-1)\), \(g_n(f_{n-1}) = g_{n-1}\) is defined as follows.
 * For \(f_{n-2} \in M(n-2)\), \(g_{n-1}(f_{n-2}) = g_{n-2}\) is defined as follows.
 * For \(f_0 \in M(0)\), \(g_1(f_0) = g_0\) is defined as follows.
 * \(g_0 = (..((f_n^{f_0}f_{n-1})f_{n-2})...f_1)f_0\)
 * \(g_0 = (..((f_n^{f_0}f_{n-1})f_{n-2})...f_1)f_0\)

Therefore, by denoting \(f_1\) as \(f\) and \(f_0\) as \(x\), \begin{eqnarray*} f(x) & = & {x}^{x} \\ (m(2)f)(x) & = & ({f}^{x})(x) \\ (..((m(n+1)f_n)f_{n-1})...f_1)(x) & = & ((..({f_n}^{f_0}{f_{n-1}})...f_2)f)(x) \end{eqnarray*}

And it works out like this.

\begin{eqnarray*} m(4) m(3) m(2) m(1)(x) & = & m(3)^x m(2) m(1)(x) \approx f_{\omega^\omega}(x) \\ m(3) [m(4) m(3)] m(2) m(1)(x) & = & [m(3)^{x} m(2)]^x m(1)(x) \approx f_{\omega^{\omega+1}}(x) \\ m(3)^2 [m(4) m(3)] m(2) m(1)(x) & \approx & f_{\omega^{\omega+2}}(x) \\ m(3)^a [m(4) m(3)] m(2) m(1)(x) & \approx & f_{\omega^{\omega+a}}(x) \\ [ m(4) m(3) ] ^2 m(2) m(1)(x) & = & [ m(3)^x ] [ m(3)^x m(2)] m(1)(x) \approx f_{\omega^{\omega \times 2} } (x) \\ [ m(4) m(3) ] ^3 m(2) m(1)(x) & \approx & f_{\omega^{\omega \times 3} } (x) \\ [ m(4) m(3) ] ^a m(2) m(1)(x) & \approx & f_{\omega^{\omega \times a} } (x) \\ [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2} } (x) \\ m(3) \bigl[ m(4)^2 m(3) ]  m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+1} } (x) \\ [ m(4) m(3) ] [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+\omega} } (x) \\ [ m(4) m(3) ] ^2 [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+\omega \times 2} } (x) \\ [ m(4) m(3) ] ^3 [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+\omega \times 3} } (x) \\ [ m(4)^2 m(3) ] ^2 m(2) m(1)(x) & \approx & f_{\omega^{\omega^2 \times 2} } (x) \\ [ m(4)^2 m(3) ] ^3 m(2) m(1)(x) & \approx & f_{\omega^{\omega^2 \times 3} } (x) \\ m(4)^3 m(3) m(2) m(1)(x) & \approx & f_{\omega^{\omega^3} } (x) \\ m(4)^4 m(3) m(2) m(1)(x) & \approx & f_{\omega^{\omega^4} } (x) \\ m(5) m(4) m(3) m(2) m(1)(x) & \approx & f_{\omega^{\omega^\omega} } \end{eqnarray*}

It was shown that \begin{eqnarray*} m(3) m(2) m(1)(x) & = & f_{\omega}(x) \\ m(4) m(3) m(2) m(1)(x) & = & f_{\omega^\omega}(x) \\ m(5) m(4) m(3) m(2) m(1)(x) & = & f_{\omega^{\omega^\omega}}(x) \end{eqnarray*} and similarly, \begin{eqnarray*} m(6) m(5) m(4) m(3) m(2) m(1)(x) & = & f_{\omega^4}(x) \\ m(7) m(6) m(5) m(4) m(3) m(2) m(1)(x) & = & f_{\omega^5}(x) \\ & … & \\ ((..((m(x)m(x-1))m(x-2))...m(2))m(1))(x) & = & f_{\varepsilon_0}(x) \end{eqnarray*}

Structure of \(m(n)\) is like Kirby-Paris hydra and corresponding ordinal. In the equation \[[ m(4) m(3) ] ^3 [ m(4)^2 m(3) ] m(2) m(1)(x) \approx f_{\omega^{\omega^2+\omega \times 3} } (x)\]

\([m(4) m(3)]^3\) corresponds to \(\omega \times 3\), \(m(4)^2\) corresponds to 2, and \(m(4)^2 m(3)\) corresponds to \(\omega^2\).



In the figure (Kirby and Paris, 1982), the leftmost top node is \(m(4)\), because it is the 4th node from the root. The node one segment below it is \(m(3)\), and it has 3 \(m(4)\) above, and therefore it is \(m(4)^3 m(3)\). The node below it is \(m(2)\) node with \(m(4)^3 m(3)\) and \(m(5)m(4)m(3)\), and therefore \([m(4)^3 m(3)][m(5)m(4)m(3)]m(2)\). The root node is \([m(3)[m(4)^2 m(3)]m(2)]\;[[m(4)^3 m(3)][m(5)m(4)m(3)]m(2)]\;m(1)\).

\(m(n)\) map can thus be correlated with the hydra tree, which is correlated with ordinal of Cantor normal form (see the cited paper of Kirby and Paris, 1982). Therefore it has the growth rate of \(f_{\varepsilon_0}(x)\).