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This is my first serious attempt at googology. I hope I didn't screw up too badly ^.^

The Basic Idea of The W-function

The W-function takes the following form:

\[W(X_n,X_{n-1},...,X_2,X_1)\]

Rule 1. If there is only one entry:

\[W(X) = 2 ^ X\]

Rule 2. If there are two or more entries, with no leading zeroes:

\[W(X_n,X_{n-1},...,X_2,X_1) = W(X_n-1,\underbrace{W(X_n-1,X_{n-1},...,X_1,X_0),...,W(X_n-1,X_{n-1},...,X_2,X_1)}_{n-1})\]

This is less confusing when looking at the 2-entry instance:

\[W(X_2,X_1) = W(X_2-1,W(X_2-1,X_1))\]

Rule 3. Leading zeroes are deleted if they exist:

\[W(0, X_n,X_{n-1},...,X_1) = W(X_n,X_{n-1},...,X_1)\]

First Extension: WW-function

Let

\[WW(X) = W(\underbrace{X,X,X...,X,X,X}_X)\]

\[WW^n(X) = \underbrace{WW(WW(WW...(WW(X))...))}_n\]

Second Extension: WW[n]Y-Function

Let

\[WW[1]Y(X) = WW^{WW(X)}(X)\]

\[WW[n]Y(X) = WW[n-1]Y^{WW[n-1]Y(X)}(X)\]

Results

W(0) to Aecronul

One-entry level

\[W(0) = 2^0 = 1\]

\[W(1) = 2^1 = 2\]

\[W(2) = 2^2 = 4\]

\[W(3) = 2^3 = 8\]

\[W(5) = 2^5 = 32\]

\[W(10) = 2^{10} = 1024\]

\[W(16) = 2^{16} = 65536\]

\[W(20) = 2^{20} = 1048576\]

Two-entry level

\[W(1,0) = 2\]

\[W(1,1) = 4 = 2^2\]

\[W(1,2) = 16 = 2^{2^2}\]

\[W(2,0) = 16 = 2^{2^2}\]

\[W(1,3) = 256 = 2^{2^3}\]

\[W(1,4) = 65536 = 2^{2^{2^2}}\]

\[W(2,1) = 65536 = 2^{2^{2^2}}\]

\[W(2,2) = {}^52\]

\[W(3,0) = {}^72\]

\[W(4,0) = {}^{15}2\]

Multiple-entry level

\[W(1,0,0) = 4\]

\[W(1,0,1) = 16\]

\[W(1,1,0) = {}^52\]

\[W(1,0,2) = {}^{18}2 = \mbox{Aecronul}\]

Aecronul to Exronaxul

Multiple-entry level

\[W(1,1,1) = {}^{18}2 = \mbox{Aecronul}\]

\[W(3,0,0) = \mbox{Aecraxul}\]

\[W(3,3,3) = \mbox{Aecronaxul}\]

\[W(1,1,1,1) = \mbox{Erlfuronul}\]

\[W(4,0,0,0) = \mbox{Erlfuraxul}\]

\[W(4,4,4,4) = \mbox{Erlfuronaxul}\]

\[W(1,1,1,1,1) = \mbox{Emtwonul}\]

\[W(5,0,0,0,0) = \mbox{Emtwaxul}\]

\[W(5,5,5,5,5) = \mbox{Emtwonaxul}\]

WW level

\[WW(5) = \mbox{Emtwonaxul}\]

\[WW(6) = \mbox{Musthonaxul}\]

\[WW(7) = \mbox{Ouvcosonaxul}\]

\[WW(8) = \mbox{Exronaxul}\]

Exronaxul to Exronaxulplex

WW level

\[WW(8) = \mbox{Exronaxul}\]

\[WW(9) = \mbox{Faezonaxul}\]

\[WW(10) = \mbox{Seritonaxul}\]

\[WW(100) = \mbox{Googonaxul}\]

\[WW(WW(8)) = \mbox{Drexronaxul}\]

\[WW(WW(WW(8))) = \mbox{Trexronaxul}\]

\[WW(WW(WW(WW(8)))) = \mbox{Tetrexronaxul}\]

Iterated WW level

\[WW^4(8) = \mbox{Tetrexronaxul}\]

\[WW^5(8) = \mbox{Pentexronaxul}\]

\[WW^6(8) = \mbox{Hextexronaxul}\]

\[WW^7(8) = \mbox{Heptexronaxul}\]

\[WW^8(8) = \mbox{Octexronaxul}\]

\[WW^9(8) = \mbox{Ennexronaxul}\]

\[WW^{WW(8)}(8) = \mbox{Exronaxulplex}\]

Exronaxulplex to Forcexronaxul

WW[n]Y level

\[WW[1]Y(8) = \mbox{Exronaxulplex}\]

\[WW[1]Y(WW(8)) = \mbox{Drexronaxulplex}\]

\[WW[1]Y(WW^2(8)) = \mbox{Trexronaxulplex}\]

\[WW[2]Y(8) = \mbox{Exronaxuldrex}\]

\[WW[2]Y(WW(8)) = \mbox{Drexronaxuldrex}\]

\[WW[3]Y(8) = \mbox{Exronaxultrex}\]

\[WW[4]Y(8) = \mbox{Exronaxultetrex}\]

\[WW[5]Y(8) = \mbox{Exronaxulpentex}\]

\[WW[6]Y(8) = \mbox{Exronaxulhextex}\]

\[WW[7]Y(8) = \mbox{Exronaxulheptex}\]

\[WW[8]Y(Y) = \mbox{Exronaxuloctex}\]

\[WW[9]Y(Y) = \mbox{Exronaxulennex}\]

\[WW[WW(8)]Y(8) = \mbox{Dypexronaxul}\]

\[WW[WW^2(8))]Y(8) = \mbox{Dydrexronaxul}\]

\[WW[WW[1]Y(8)]Y(8) = \mbox{Dypexronaxulplex}\]

Array WW level

\[WW(3,8)\] = Octexronaxul

\[WW(3,0,0)\] = Forcaecraxul

\[WW(3,3,3)\] = Forcaecronaxul

\[WW(4,0,0,0)\] = Forcerlfuraxul

\[WW(4,4,4,4)\] = Forcerlfuronaxul

\[WW(5,5,5,5,5)\] = Forcemtwonaxul

WWW level

\[WWW(6)\] = Forcemusthonaxul

\[WWW(7)\] = Forcouvcosonaxul

\[WWW(8)\] = Forcexronaxul

Forcexronaxul to Exriforcexronaxul

WWW level

\[WWW(8)\] = Forcexronaxul

\[WWW(WW(8))\] = Forcedrexronaxul

\[WWW(WW[1]Y(8))\] = Forcexronaxulplex

\[WWW(WW[1]Y(WW(8)))\] = Forcedrexronaxulplex

\[WWW(WWW(8))\] = Dryrforcexronaxul

\[WWW(WWW(WWW(8)))\] = Tryrforcexronaxul

\[WWW(WWW(WWW(WWW(8))))\] = Tetryrforcexronaxul

Array WWW level

\[WWW(2,8)\] = Tetryrforcexronaxul

\[WWW(3,8)\] = Octyrforcexronaxul

\[WWW(3,3,3)\] = Forceforcaecronaxul

\[WWW(8,8,8,8,8,8,8,8)\] = Forceforcexronaxul

Subscript-W Level

\[W(4)\_(8)\] = Forceforcexronaxul

\[W(4)\_(WW(8))\] = Forceforcedrexronaxul

\[W(4)\_(WWW(8))\] = Forcedryrforcexronaxul

\[W(4)\_(W(4)\_(8))\] = Dryrforceforcexronaxul

\[W(4)\_(W(4)\_(WWW(8)))\] = Dryrforcedryrforcexronaxul

\[W(4)\_(2,8)\] = Tetryrforceforcexronaxul

\[W(4)\_(3,8)\] = Octyrforceforcexronaxul

\[W(5)\_(3)\] = Emtwiforcaecronaxul

\[W(5)\_(8)\] = Emtwiforcexronaxul

\[W(5)\_(WW(8))\] = Emtwiforcedrexronaxul

\[W(5)\_(WWW(8))\] = Forceforcedryrforcexronaxul

\[W(5)\_(W(4)\_(8))\] = Forcedryrforceforcexronaxul

\[W(5)\_(W(5)\_(8))\] = Emtwidryrforcexronaxul

\[W(6)\_(8)\] = Musthiforcexronaxul

\[W(6)\_(WW(8))\] = Musthiforcedrexronaxul

\[W(6)\_(WWW(8))\] = Emtwiforcedryrforcexronaxul

\[W(6)\_(W(4)\_(8))\] = Forceforcedryrforceforcexronaxul

\[W(6)\_(W(5)\_(8))\] = Forcemtwidryrforcexronaxul

\[W(6)\_(W(6)\_(8))\] = Musthidryrforcexronaxul

\[W(7)\_(8)\] = Ouvcosiforcexronaxul

\[W(8)\_(8)\] = Exriforcexronaxul

Exriforcexronaxul to Exriaxstexronaxul

Subscript-W level

\[W(8)\_(8) = Exriforcexronaxul\]

\[W(256)\_(8) = Saragicijziforcexronaxul\]

Array-Subscript-W level

\[W(1,1)\_(8)\] = Saragicijziforcexronaxul

\[W(3,3,3)\_(3)\] = Axstaecronaxul

\[W(8,8,8,8,8,8,8,8)\_(8)\] = Axstexronaxul

Tiered-Subscript-W level

\[W(2)\_(8)\_(8)\] = Axstexronaxul

\[W(2)\_(8)\_(WW(8))\] = Axstdrexronaxul

\[W(2)\_(8)\_(WWW(8))\] = Axstforcexronaxul

\[W(2)\_(8)\_(W(2)\_(8)\_(8))\] = Dryraxstexronaxul

\[W(2)\_(W(2)\_(8)\_(8))\_(8)\] = Draxstexronaxul

\[W(2)\_(W(2)\_(8)\_(8))\_(W(2)\_(8)\_(8))\] = Dydraxstexronaxul

\[W(2)\_(W(2)\_(8)\_(8))\_(W(2)\_(W(2)\_(8)\_(8))\_(8))\] = Dryrdraxstexronaxul

\[W(2)\_(W(2)\_(8)\_(W(2)\_(8)\_(8)))\_(W(2)\_(8)\_(8))\] = Drdryraxstexronaxul

\[W(3)\_(8)\_(8)\] = Axstaxstexronaxul

\[W(3)\_(8)\_(W(2)\_(8)\_(8)\)\] = Axstdryraxstexronaxul

\[W(3)\_(8)\_(W(3)\_(8)\_(8))\] = Dryraxstaxstexronaxul

\[W(3)\_(W(3)\_(8)\_(8))\_(W(3)\_(8)\_(8))\] = Dydraxstaxstexronaxul

\[W(3)\_(W(2)\_(8)\_(8)\))\_(8)\] = Axstdraxstexronaxul

\[W(3)\_(W(2)\_(8)\_(8)\))\_(W(2)\_(8)\_(8))\] = Axstdydraxstexronaxul

\[W(3)\_(W(2)\_(8)\_(8)\))\_(W(3)\_(8)\_(8))\] = Dryraxstdraxstexronaxul

\[W(3)\_(W(3)\_(8)\_(8))\_(8)\] = Draxstaxstexronaxul

\[W(3)\_(W(3)\_(8)\_(8))\_(W(2)\_(8)\_(8))\] = Draxstdryraxstexronaxul

\[W(3)\_(W(3)\_(8)\_(8))\_(W(3)\_(8)\_(8))\] = Dydraxstaxstexronaxul

\[W(4)\_(8)\_(8)\] = Axstaxstaxstexronaxul

\[W(4)\_(8)\_(W(1)\_(8)\_(8))\] = Axstaxstaxstexriforcexronaxul

\[W(4)\_(8)\_(W(2)\_(8)\_(8))\] = Axstaxstdryraxstxronaxul

\[W(4)\_(8)\_(W(3)\_(8)\_(8))\] = Axstdryraxstaxstxronaxul

\[W(4)\_(8)\_(W(4)\_(8)\_(8))\] = Dryraxstaxstaxstxronaxul

\[W(4)\_(W(2)\_(8)\_(8))\_(8)\] = Axstaxstdraxstxronaxul

\[W(4)\_(W(3)\_(8)\_(8))\_(8)\] = Axstdraxstaxstxronaxul

\[W(4)\_(W(3)\_(8)\_(8))\_(8)\] = Axstdraxstaxstxronaxul

\[W(4)\_(W(4)\_(8)\_(8))\_(8)\] = Draxstdraxstaxstxronaxul

\[W(5)\_(8)\_(8)\] = Emtwiaxstexronaxul

\[W(5)\_(8)\_(W(2)\_(8)\_(8))\] = Axstaxstaxstdryraxstexronaxul

\[W(5)\_(8)\_(W(5)\_(8)\_(8))\] = Dryremtwiaxstexronaxul

\[W(5)\_(W(2)\_(8)\_(8))\_(8)\] = Axstaxstaxstdraxstexronaxul

\[W(5)\_(W(5)\_(8)\_(8))\_(8)\] = Dremtwiaxstexronaxul

\[W(5)\_(W(5)\_(8)\_(8))\_(W(5)\_(8)\_(8))\] = Dydremtwiaxstexronaxul

\[W(6)\_(8)\_(8)\] = Musthiaxstexronaxul

\[W(6)\_(8)\_(W(2)\_(8)\_(8))\] = Emtwiaxstdryraxstexronaxul

\[W(6)\_(8)\_(W(5)\_(8)\_(8))\] = Dryraxstemtwiaxstexronaxul

\[W(6)\_(8)\_(W(6)\_(8)\_(8))\] = Dryrmusthiaxstexronaxul

\[W(6)\_(W(2)\_(8)\_(8))\_(8)\] = Emtwiaxstdraxstexronaxul

\[W(6)\_(W(2)\_(8)\_(8))\_(W(2)\_(8)\_(8))\] = Emtwiaxstdydraxstexronaxul

\[W(6)\_(W(5)\_(8)\_(8))\_(8)\] = Draxstemtwiaxstexronaxul

\[W(6)\_(W(5)\_(8)\_(8))\_(W(5)\_(8)\_(8))\] = Dydraxstemtwiaxstexronaxul

\[W(6)\_((6)\_(8)\_(8))\_(8)\] = Dremusthiaxstexronaxul

\[W(7)\_(8)\_(8)\] = Ouvcosiaxstexronaxul

\[W(8)\_(8)\_(8)\] = Exriaxstexronaxul

More Extensions

Special thanks to Alejandro Magno for help with the 3rd through 7th extensions!

Third Extension: Array WW-function

The WW-function can be extended to handle arrays in the following manner:

\[WW(X_n,X_{n-1},...,X_2,X_1) = WW(X_n-1,\underbrace{WW(X_n-1,X_{n-1},...,X_2,X_1),WW(X_n-1,X_{n-1},...,X_2,X_1),...,WW(X_n-1,X_{n-1},...,X_2,X_1)}_{n-1})\]

\[WW(0,X_n,X_{n-1},...,X_2,X_1) = WW(0,X_n,X_{n-1},...,X_2,X_1)\]

In much the same way as the single W-function.

As a result, if the starting arrays are sufficiently large:

\[WW(X_n,X_{n-1},...,X_2,X_1) > WW(W(X_n,X_{n-1},...,X_2,X_1))\]

Example:

\[WW(1,1) = WW(0,WW(0,1))\]

\[ = WW(W(1))\]

\[ = WW(2))\]

\[ = W(2,2)\]

\[ = ^52\]

Fourth Extension: Subscript Multiple-W-function

We can define:

\[WWW(X) = WW(\underbrace{X,X,X,...,X,X,X}_X)\]

\[WWW(X_n,X_{n-1},...,X_2,X_1) = WWW(X_n-1,\underbrace{WWW(X_n-1,X_{n-1},...,X_2,X_1),WWW(X_n-1,X_{n-1},...,X_2,X_1),...,WWW(X_n-1,X_{n-1},...,X_2,X_1)}_{n-1})\]

\[WWW(0,X_n,X_{n-1},...,X_2,X_1) = WWW(X_n,X_{n-1},...,X_2,X_1)\]

Further Ws can then be added:

\[\underbrace{WWW...WWW}_Z(X) = \underbrace{WWW...WWW}_{Z-1}(X)(\underbrace{X,X,X,...,X,X,X}_X)\]

\[\underbrace{WWW...WWW}_Z(X_n,X_{n-1},...,X_2,X_1) = \underbrace{WWW...WWW}_Z(X_n-1,\underbrace{\underbrace{WWW...WWW}_Z(X_n-1,X_{n-1},...,X_2,X_1),\underbrace{WWW...WWW}_Z(X_n-1,X_{n-1},...,X_2,X_1),...,\underbrace{WWW...WWW}_Z(X_n-1,X_{n-1},...,X_2,X_1)}_{n-1})\]

\[\underbrace{WWW...WWW}_Z(0,X_n,X_{n-1},...,X_2,X_1) = \underbrace{WWW...WWW}_Z(X_n,X_{n-1},...,X_2,X_1)\]

To make functions with many Ws more compact, the following notation is introduced:

\[W(Z)\_(X_n,X_{n-1},...,X_2,X_1) = \underbrace{WWW...WWW}_Z(X_n,X_{n-1},...,X_2,X_1)\]

\[W(Z)\_(X) = W(Z-1)\_(\underbrace{X,X,X,...,X,X,X}_X)\]

\[W(Z)\_(X_n,X_{n-1},...,X_2,X_1) = W(Z)\_(X_n-1,\underbrace{W(Z)\_(X_n-1,X_{n-1},...,X_2,X_1),W(Z)\_(X_n-1,X_{n-1},...,X_2,X_1),...,W(Z)\_(X_n-1,X_{n-1},...,X_2,X_1)}_{n-1})\]

\[W(Z)\_(0,X_n,X_{n-1},...,X_2,X_1) = W(Z)\_(X_n,X_{n-1},...,X_2,X_1)\]

Fifth Extension: Array-subscript Multiple-W-function

From now on, the X array is not particularly relevant, so it will be abbreviated &_X.

The subscript W-function can be extended to take arrays as subscripts as follows:

\[W(Z_m,Z_{m-1},...,Z_2,Z_1)\_(\&_X) = W(Z_m-1,\underbrace{W(Z_m-1,Z_{m-1},...,Z_2,Z_1)\_(\&_X),...,W(Z_m-1,Z_{m-1},...,Z_2,Z_1)\_(\&_X)}_{m-1})\_(\&_X)\]

\[W(0,Z_m,Z_{m-1},...,Z_2,Z_1)\_(\&_X) = W(Z_m,Z_{m-1},...,Z_2,Z_1)\_(\&_X)\]

Sixth Extension: Two-tiered-subscript Multiple-W-function

The subscript W-function can be extended to 2 subscripts as follows:

\[W(1)\_(\&_{Z_1})\_(\&_X) = W(\&_{Z_1})\_(\&_X)\]

\[W(Z_2)\_(Z_1)\_(\&_X) = W(Z_2-1)\_(\underbrace{Z_1,Z_1,Z_1,...,Z_1,Z_1,Z_1}_{Z_1})\_(\&_X)\]

\[W(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}) = (Z_{2,m_2}-1,\underbrace{(Z_{2,m_2}-1,Z_{2,m_2-1},...,Z_{2,2},Z_{2,1})\_(Z_1)\_(\&_X),...,(Z_{2,m_2}-1,Z_{2,m_2-1},...,Z_{2,2},Z_{2,1})\_(Z_1)\_(\&_X)}_{m_2-1})\_W(Z_1)\_(\&_{X})\]

\[W(Z_2)\_(\&_{Z_1})\_(\&_{X}) = W(Z_2)\_(Z_{1,m_1}-1,\underbrace{W(Z_2)\_(Z_{1,m_1}-1,Z_{1,m_1-1},...,Z_{1,2},Z_{1,1})\_(\&_X)}_{m_1-1})\_(\&_{X})\]

The arrays must always be solved from left to right.

Seventh Extension: Multiple-tiered-subscript Multiple-W-function

The two-tiered notation can be further extended to more tiers in the following manner:

\[W(1)\_(\&_{Z_p})\_(\&_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}) = W(\&_{Z_p})\_(\&_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X})\]

\[W(Z_p)\_(Z_{p-1})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}) = W(Z_p-1)\_(\underbrace{Z_{p-1},Z_{p-1},Z_{p-1},...,Z_{p-1},Z_{p-1},Z_{p-1}}_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X})\]

\[W(Z_p)\_(\&_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}) = W(Z_p)\_(Z_{{p-1},m_{p-1}}-1,\underbrace{W(Z_p)\_(Z_{{p-1},m_{p-1}}-1,Z_{p,m_p-1},...,Z_{{p-1},2},Z_{{p-1},1})\_(\&_{Z_{p-2}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}),...,W(Z_p)\_(Z_{{p-1},m_{p-1}}-1,Z_{p,m_p-1},...,Z_{{p-1},2},Z_{{p-1},1})\_(\&_{Z_{p-2}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X})}_{m_{p-1}-1})\_(Z_{p-2})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X})\]

\[W(\&_{Z_p})\_(\&_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}) = W(Z_{p,m_p}-1,\underbrace{W(Z_{p,m_p}-1,Z_{p,m_p-1},...,Z_{p,2},Z_{p,1})\_(\&_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X}),...,W(Z_{p,m_p}-1,Z_{p,m_p-1},...,Z_{p,2},Z_{p,1})\_(\&_{Z_{p-1}})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X})}_{m_p-1})\_(Z_{p-1})\_...\_(\&_{Z_2})\_(\&_{Z_1})\_(\&_{X})\]

Again, the arrays need to be solved from left to right. You can't solve any other than the leftmost two.

Eighth Extension: Second-order-subscript W-function

Sorry, screwed up this section's equations first time around. They're getting really nasty, so here's a summary of all the rules in words so nobody gets lost:

-- If there's a single number and no subscript, take 2 to the power of that number.

-- If there's a zero in front of an array, it is deleted.

-- Array Rule: If the highest-tier subscript is an array, or it's a single number but the next-highest-tier subscript is an array, the first entry is reduced by 1, and the other entries are replaced with the entire function with the first entry in that array reduced by 1.

-- If the highest-tier subscript is 1, it is deleted.

-- If there's a single number and a single-number-subscript, the subscript is reduced by 1 and the other is replaced with an array of itself with a size of itself.

-- If there's a single-number subscript and a single-number subscript of the next tier down, the highest-tier subscript is reduced by 1 and the next tier subscript is replaced with an array of itself with a size of itself.

With the introduction of second-order subscripts, I add:

-- If the second-order subscript is 1, it is deleted:

\[W(1)\_\_(Z)\_(X) = W(Z)\_(X)\]

-- If the second-order subscript is a single number with a single tier and so is the first-order subscript, the second-order subscript is reduced by 1 and the first-order subscript is replaced with itself with a number of tiers of first-order subscripts equal to itself minus one, so that there are that number of levels when including the base:

\[W(Z_2)\_\_(Z_1)\_(X) = W(Z_2-1)\_\_\underbrace{(Z_1)\_(Z_1)\_...\_(Z_1)\_(Z_1)}_(Z_1)\_{X}\]

-- When the second-order subscript is an array and/or has multiple tiers, the array rule proceeds as normal:

(Not even going to attempt this one, sorry...)

-- When the second-order subscript is a single number with a single tier but the first-order subscript isn't, the array rule proceeds as normal, making sure to include the second-order subscript as well where appropriate: (The mistake I made in the first version of this section was failing to include it, counting the stack of first-order subscripts as a separate function.)

(This would be too long for most LaTeX editors anyway, sigh...)

Ninth Extension: Multiple-tiered Second-order-subscript W-function

-- If the highest-tier second-order subscript is 1, it is deleted:

\[W(1)\_\_(Z^2_1)\_\_(Z)\_(X) = W(Z^2_1)\_\_(Z)\_(X)\]

-- If the highest-tier and second-highest-tier second-order subscripts are both single numbers, the highest-tier second-order subscript is reduced by 1 and the second-highest-tier second-order subscript is replaced with itself with a number of first-order subscripts equal to itself minus one:

\[W(Z^2_2)\_\_(Z^2_1)\_\_(Z)\_(X) = W(Z_^2_2-1)\_\_\underbrace{(Z^2_1)\_(Z^2_1)\_...\_(Z^2_1)\_(Z^2_1)}_(Z^2_1)\_{X}\_\_(Z)\_(X)\]

The array rule equations are stupid crazy so I won't bother... Just one more extension and then I can stop for a while.

Tenth Extension: Higher-order-subscript W-function

Let

\[\__V\_\]

indicate an Vth-order subscript.

-- When the highest-order subscript is 1, it is deleted:

\[W(1)\__{V+1}\_(Z^{V})\__{V}\_(Z^{V-1})\__{V-1}\_...\__{3}\_(Z^2)\_\_(Z^1)\_(X)\ = W(Z^{V})\__{V}\_(Z^{V-1})\__{V-1}\_...\__{3}\_(Z^2)\_\_(Z^1)\_(X)\]

-- When the highest-order and second-highest-order subscripts are both 1, the highest-order subscript is reduced by 1 and the second-highest-order subscript is replaced by itself with a number of second-highest-order tiers equal to itself minus 1:

\[W(Z^{V})\__{V}\_(Z^{V-1})\__{V-1}\_...\__{3}\_(Z^2)\_\_(Z^1)\_(X) = W(Z^{V}-1)\__{V}\_{(Z^{V-1})\__{V-1}\_...\__{V-1}\_(Z^{V-1})\__{V-1}}_{V-1}\__{V-2}\_...\__{3}\_(Z^2)\_\_(Z^1)\_(X)\]

-- Array rule works like it always does.

Subscripts require every lower order or tier to also have a non-1-subscript to function effectively, due to the fact that WW(1) = W(1), W(Z)_(1) = 1, W(Z)\__V\_(1) = 1, and so on, which means that any subscripts up to and including a 1-subscript do nothing. The double-comma notation is introduced to make high-order and high-tier subscripts easier to write:

X with X-1 tiers of subscripts equal to X:

\[W(1,,X) = W\underbrace{(X)_(X)_..._(X)_(X)}_X\]

X with X-1 orders of subscripts equal to X:

\[W(2,,X) = W(X)\__{X-1}\_(X)\__{X-2}\_(X)\__{X-3}\_...\__3\_(X)\_\_(X)\_(X)]

Eleventh Extension: Dimensional-subscript W-function

Dimensions are to orders what orders are to tiers. Dimensional subscripts are notated:

[\\underbar{_{U}}\]

Where U is the dimension of the subscript.

-- When the highest-dimension subscript is 1, it is deleted:

\[W(1)\underbar{_{U+1}}(V^U)\underbar{_U}(V^{U-1})\underbar{_{U-1}}(V^{U-2})\underbar{_{U-2}}...\underbar{_{3}}(V^2)\underbar{_{2}}(Z)_(X) = W(V^U)\underbar{_U}(V^{U-1})\underbar{_{U-1}}(V^{U-2})\underbar{_{U-2}}...\underbar{_{3}}(V^2)\underbar{_{2}}(Z)_(X)\]

-- When the highest-dimension subscript and second-highest-dimension subscript are both single numbers, reduce the highest-dimension subscript by 1 and the replace the second-highest-dimension subscript with itself with a number of second-highest-dimensional orders equal to itself minus 1:

\[W(V^U)\underbar{_U}(V^{U-1})\underbar{_{U-1}}(V^{U-2})\underbar{_{U-2}}...\underbar{_{3}}(V^2)\underbar{_{2}}(Z)_(X) = W(V^U-1)\underbar{_U}(V^{U-1})\underbar{_{U-1}}_{V^{U-1}-1}\underbar{_{U-1}}(V^{U-1})\underbar{_{U-1}}_{V^{U-1}-2}\underbar{_{U-1}}...\underbar{_{U-1}}\underbar{_{U-1}}(V^{U-1})\underbar{_{U-1}}(V^{U-1})(V^{U-2})\underbar{_{U-2}}...\underbar{_{3}}(V^2)\underbar{_{2}}(Z)_(X)\]

In double-comma notation, X with X-1 dimensions of subscripts equal to X:

\[W(3,,X) = W(X)\underbar{_{X-1}}(X)\underbar{_{X-2}}(X)\underbar{_{X-3}}...\underbar{_{3}}(X)\underbar{_{X-2}}(X)\_(X)\]

EDIT: LaTeX hates me, but I think you can figure out what's going on.

Twelfth Extension: Double-comma W-function

If tiers are the first "category", orders are the second, and dimensions are the third, then:

-- The highest category is called the primary category and the second-highest category is called the subcategory

-- If the primary category subscript is equal to 1, it is deleted.

-- If the highest primary category and second-highest primary category subscript are both single numbers, then the highest primary category subscript is reduced by 1 and the second-highest primary category subscript is replaced with itself with a number of subcategory subscripts equal to itself minus 1.

\[W(A,,X)\]

is X with X-1 category-A subscripts equal to X.

Thirteenth Extension: Hypercategory Array Double-comma W-function

(Coming soon!)

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