User blog:Nayuta Ito/Another Extension of Fish Numbers

First, I will write the definition of S map.

(Quote from Japanese Googology Wiki, translated by me)

[1] S map, a map between pairs of a natural number and a function, is defined like this.

$$S(m,f(x)) = (g(m),g(x))$$

And here's the definition of \(g(x)\).

$$B(0,n) =  f(n) $$

$$B(m+1,0) =  B(m, 1)$$

$$B(m+1,n+1) = B(m, B(m+1, n))$$

$$g(x) =  B(x,x)$$

You can also write like this focusing on the function: \[Sf(x) = g(x)\]

(Unquote)

And the definition of SS map is this, using S map:

$$SS(m,f,A)=(A^{f(m)}f(m),A^{f(m)}f,A^{f(m)})$$

(A is a map. Many people use S, but I used A for disambiguation.)

Now it's getting crazy. Here's some definition:

A (non-meta)map is a map between sets of a natural number and a function. (e.g. S map)

A meta-map is a map between sets of a natural number, a function, and a map. (e.g. SS map)

A meta-meta-map is a map between sets of a natural number, a function, a map, and a meta-map.

A meta-meta-meta-map is a map between sets of a natural number, a function, a map, a meta-map, and a meta-meta-map.

Get the point? This pattern continues forever.

Now it's getting crazier.

Sn map is a metan-1map that is defined like this:

$$S_n(m,f,A_0,A_1,\cdots A_{n-2})=(A_{n-2}^{f(m)}f(m), A_{n-2}^{f(m)}f, A_{n-2}^{f(m)}A_0, A_{n-2}^{f(m)}A_1, \cdots A_{n-2}^{f(m)}A_{n-3}, A_{n-2}^{f(m)})$$

,where A_m is a metammap.

Now it's getting craziest.

S+ map