Map hierarchy

${\textstyle n<\omega}$

$M_0(n)$ Is the same as $f_{0}(n)$ in the fast growing hierarchy. Dont know what that is? Its just $n+1$.

Anyways, the $M$ hierarchy maps (nearly) every function in googology wiki's list of functions (as of 6/1/2024)

$M^m_0(n)=n+m$. Note $n+m$ isnt expressible in $M_a(n)$ without using iterations.

$M_1(n)=2n$

$M_2(n)=n^n, M^2_2(m)=M_2(M_2(n))=M_2(n^n)=(n^n)^(n^n)$

$M_3(n)=n!, M^2_3(n)=M_3(n!)=(n!)!$

$M_4(n)=n$=1!*2!*3!...$ all the way to $n!$.

$M_5(n)=T(n)=n!n$. Dont want to go to Torian's page? Its basically recapped as:

$x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)$

$M_6(n)=H(n)$ using hyperfactorials.

$M_7(n)=L(n)$ using the latin square.

$M_8(n)=n \uparrow n \uparrow n$

$M_9(n)=n \backslash$.

$M_{10}(n)=$googol-n-ple-n ($n \uparrow (2n) \uparrow n$)

$M_{11}(n)=\theta(n)$ using Bop-counting function.

$M_{12}(n)=n_n$ using exponential factorials.

We can see this is slower than FGH, as FGH reaches tetrational level at $f_{3}(n)$, but we reach that in $M_{13}(n)$, althought its not a approximation and EXACTLY tetration.

(im lazy to fix the errors in the index so pretend i shifted everything after this point by one)

$M_{13}(n)=ban(n)=n \uparrow^{2}n+1$

$M_{14}(n)=expostfacto(n)=n\uparrow expostfacto(n-1)!$ where $expostfacto(1)=1$

$M_{15}(n)=n$$ using pickover's definition.

$M_{16}=n!2$ using tetrofactorials.

$M_{17}(n)=n\uparrow^{3} n$

$M_{18}(n)=rban(n)=n\uparrow n \uparrow^{3} n$

The Map Hierarchy gives us a wow at $M_{19}(n)=f_4(n)$ or the wow function.

$M_{20}(n)=$Circle(n)

Factorials are back with $M_{21}(n)= n[5]!$ or a Pentatorial.

If $29 >a>20$, $M_a(n)=n\uparrow^{a-17}n$

$M_{29}(n)=g(n)$ using Weak goodstein, where we reach the level of $f_{\omega}(n)$

$M_{30}(n)=|T[n]|$, The finite ordered tree problem.

$M_{31}(n)=A(n,n)$, the Ackermann function.

$M_{32}(n)= booga-n = n\uparrow^{n-2}n$

$M_{33}$ up to $M_{35}$ are (in order): Mythical tree problem, Vector reduction problem, and davenport schinzel sequence.

$M_{36}$ is both arrow notation and ackermann numbers, as $M_{36}(n) =n \uparrow^{n}n$. We will be using BEAF-related things from now on, so its $3\And n$ using Array of.

$M_{37}(n)=F_n(n,n)$ in the Sudan function.

$M_{38}(n)=n [n]$ On Steinhaus-Moser notation.

$M_{39}(n)=H^*(\underbrace{n,n,\cdots n,n}_n)$

$M_{40}(n)=$En#n

$n\psi n$ on psi notation is equal to $M_{41}(n)$

Graham's number is defined as $g_{64}$ in a sequence where $g_n = \{3,3,g_{n-1}\}, g_1 = \{3,3,4\}$, it can be defined as $M_{42}(64)$ tho!

$M_{43}(n) = mag(n)$

$M_{44}(n) = trooga(n)$

$M_{45}(n)$ Explodes the trees, as its equal to $E(n)$!

$M_{46}(n) = n\{\{1\}\}n$

$M_{47}(n) = h_n(\underbrace{n,n,\cdots n,n}_n)$. A very de(hyper)licious function!

$M_{48} = n\{\{2\}\}n$

$M_{49}(n) = n^*_{n,n}$

Next 2 functions on the hierarchy are $n\{\{3\}\}n$ and $n\{\{4\}\}n$.

$M_{52}(n) = n-ag(n)$

$M_{53}(n) =$ [n,n,n,n] in graham array notation.

The next 4 of them are just variations of explosion. (up to explodotetration}

$M_{58}(n) =$ n[n###n] using Copy Notation.

The next SEVEN are just variations of detonation (up to deconation.)

$M_{66}(n) = cg(n)$

$M_{67}(n) = H^*_n(\underbrace{n,n,\cdots n,n}_n)$

$M_{68}(n) =$ [n,n{1}n] in graham array notation.

Megotions up next. IF THERES ANOTHER VARIATION OF EXPANSION I WILL- sorry.

$M_{70}(n) = \widetilde{M}_n$

CAN WE SKIP TO THE ACTUAL GOOD STUFF? sorry, its that the next TWENTY functions of the hierarchy are variations of expansion.

$M_{91}(n) = n\rightarrow_nn$ Using a extension of chained arrow notation.

more expansion variations...

$M_{94}(n) = C(n,n,n)$ using the same extension of chained arrow notation.

more expansion variations...

$M_{98}(n) = n[\underbrace{n,n,\cdots n}_n]n$, aka the chained array notation.

$M_{99}(n) = \{\underbrace{n,n,n,n,n,n...n}_{n}\} = n\And n = X\And n$

$M_{100}(n) =$ nEn#n

$M_{101}(m) = n(m)$ Using the n(k) function.

$M_{102} = Q_{\underbrace{n,0,0,\ldots,0,0}_n}$

$M_{103}(n) = A(n,n...n)$ with n ns.

$M_{104}(n) = s(n)$

$M_{105}(n) = M(n,\underbrace{0,0,\cdots0,}_nn)$, Using the hyper-moser notation.

$M_{106}(n) = \text{Forcal}_{\underbrace{n,n,\cdots n}_n}(n)$, the graham generator.

$M_{107}(n) = n![\underbrace{n,n,\cdots n,n}_n]$

$M_{108}(n) = n\uparrow\rightarrow\uparrow n$ on matthew's function.

$M_{109}(n) = \{n,n (2) n\}$

Sadly, the next 2 \ notations are ill-defined ($\{a,b(0,1)c\}$and $\{a,b(\underbrace{0,0\ldots0,0,1}_{n})c\}$) and cannot be included in the hierarchy.

$M_{110}(n) = G(n)$ as $g(n)$ grew up.

$M_{111}(n) = n\$[[n]_n]$, aka 🤑 function

Again, another ill-defined notation (${^ba} \& n$).

$M_{112}(n) = (n,n,n,n,n,n...n\{n,n\}n)$. This isnt NaN but NaN.

$M_{113}(n) = En \#^nn$. #s were definitely cascading down the stairs.

$M_{114}(n) = \#^*(n,n,n)^* n$, a expression in pound star notation.

Its time for theorems as $M_{115}(n) = Circle(n)$, a notation by friedman.

$M_{116}(n) = m(n)$, a map becoming a map.

$M_{117}(n) = hydra(n)$. Yeah, buccholz has a hydra. Wait wrong hydra-

$M_{118}(n) = worm(n)$.

$M_{119}(n)=P^n(n)$ using PSN's definition.

$M_{120}(n) = h(g(n),n)$, using the definition of marxen.c.

$M_{121}(n) = n\{X>X>X>X...>X\}n$ in X sequence hyper exponential notation.

We have a incomplete function and one that doesnt even exist so we skip them.

$M_{122}(n) = m_1(n)$, what a fusible function!

$M_{124}(n) =$ En#^...^#n, or how i could like to call it $En\#\uparrow^n\#n$.

*skip*

$M_{125}(n) = nQ_{\underbrace{n,0,0,\ldots,0,0}_{n}}(n)$

$M_{126}(n) = tree(n)$

$M_{127}(n) = TREE(n)$

idk if its a good idea to add the next one so id skip it. I also dont think collapsing-E is a good idea, since arrays with #s in it arent fully formalized. I wont use the HSU trio, as i think they are ill defined.

$M_{128}(n) = P^n[n]$, tthhee ppaaiirr sseeqquueennccee ssyysstteemm (haha? get it? theres a pair for every le-)

Hyper primitive sequence system spits out ordinals, so its skipped.

$M_{129}(n) = g^{n}(n)$ using stage array notation.

$M_{130}(n) = SCG(n)$. Quite subcubic.

$M_{131}(n) = BH(n)$. Buccholz has a hydra. The right one this time!

We will skip 4 of the next functions because i suspect BAN is ill-defined, i already talked about HSU, already taled about hyperfactorial array notation, and this variation of stage array notation spits out ordinals.

$M_{132}(n) = (0,0,0,0...)(1,1,1,1...)[n]$, with n 0s and n 1s.

$M_{133} = D^n(n)$ using loader's function.

$M_{134}(n) =F(n,n)$, aka friedman's finite trees

$M_{135}(n) = q^n(n)$ where $q^n(n) = q(q(q(q(...n...))))$ with n iterations.

Now the map hierarchy takes a turn, it might have been really slow growing, but its now uncomputable, even if it took 137 stages to do so.

$M_{136}(n) = \Sigma(n)$

$M_{137}(n) = S(n)$, the FF function.

$M_{138}(n) = PP(n)$

$M_{139}(n) = WW(n)$

$M_{140}(n) = \Sigma_n(n)$

$M_{141}(n) = b_n(n)$

$M_{142}(n) = doodle(n)$

$M_{143}(n) = \Xi(n)$

$M_{144}(n) = \Sigma_\infty(n)$

$M_{145}(n) = f^{n}(n \uparrow^{n} n)$, The strongest function in the hierarchy that is actually assigned a function (being taken from LNGN's definition). $M^3_{145}(60)$ would be a number larger than LNGN if it was a valid googologism.

For $n>146$ (again i was too lazy to shift everything after tetration up by one, so ima roll with it, just pretend its correct, even if its correct after this point)

If 147, for example is our index, it doesnt have a assigned function, so we start defining things as FGH does (including for when $n\geq \omega$.) so the 147th function of the hierarchy is defined as n iterations of the 146th operation. Of course, diagonalization is used for ordinals so $M_\omega(146) = f^{146}(146\uparrow^{146}146)$. Can you believe this wasnt the function at all? That was just the hierarchy! The real function USES the hierarchy, the function being $M(n) = M_\omega(n)$ or the Map function, its equal to the omega-th function which just diagonalizes, so it does get outgrown by bigger ordinals. $^2M(n) = M^{M(n)}(n)$ having 2 layers, iteration on the map function works exactly as it does on the map hierarchy so our new "tetrational iterator" can be used in both. We can define "a pentational iterator", where "$_{2 \leftarrow}M(n) = M\uparrow^32(n) = M\uparrow^2M(n)(n)$ or repeated applications of the tetrational iterator. we can define a hexational iterator, heck, why not define ARRAYS for M? Ive got a BETTER idea: Why not be able to apply the functions on the M hierarchy to M itself? And folks, this is the (current) end of the map function.