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In this post, I will list all of the notations and functions Googleaarex, me and the other users  who participated in his game My number is bigger! (Googology Wiki Version) coined.

Flain() Function

n!! = (((...(((n!)!)!)!)...)!)!)!)! w/ n levels

n!!! = (((...(((n!!)!!)!!)!!)...)!!)!!)!!)!! w/ n levels

n!m = n!!!...!!! w/ m !'s

Flain(n) = n!n

Flain_0(n) = Flain(n)

Flain_1(0) = Flain(200!200)

Flain_1(n) = Flain(Flain_1(n-1))

Flain_m(0) = Flain_m-1(200!200)

Flain_m(n) = Flain_m-1(Flain_m(n-1))

Arrow() Function

^ means an up arrow, and v means a down arrow.

Arrow(n,m,0) = n*m

Arrow(n,m,1) = n^m

Arrow(n,m,2) = n^vm

Arrow(n,m,3) = n^^m

Arrow(n,m,4) = n^vvm

Arrow(n,m,5) = n^v^m

Arrow(n,m,6) = n^^vm

Arrow(n,m,7) = n^^^m

Arrow(n,m,8) = n^vvvm

Arrow(n,m,9) = n^vv^m

Arrow(n,m,10) = n^v^vm

Arrow(n,m,11) = n^v^^m

Arrow(n,m,12) = n^^vvm

Arrow(n,m,13) = n^^v^m

Arrow(n,m,14) = n^^^vm

Arrow(n,m,15) = n^^^^m

Arrow(n,m,16) = n^vvvvm

Formula

for Arrow(n,m,k), write k in binary, then replace each 1 with an up arrow, and each 0 with a down arrow.  That is the expression you put between n and m.

Extensions

Two and 1 Entry

Arrow(n,m) = Arrow(n,n,m)

Arrow(n) = Arrow(n,n)

()-Notation

Arrow(Arrow(10,10,10),Arrow(10,10,10),Arrow(10,10,10)) = a = (1)

Arrow(Arrow(a,a,a),Arrow(a,a,a),Arrow(a,a,a)) = b = (2)

Arrow(Arrow(b,b,b),Arrow(b,b,b),Arrow(b,b,b)) = c = (3)

(a) = ((1))

(a+1) = ((1)1)

((1)a) = ((1)(1))

((2)) = ((1)(1)(1)...(1)(1)(1)) w/ a (1)'s

((2)(2)) = ((2)(1)(1)(1)...(1)(1)(1)) w/ a (1)'s

((3)) = ((2)(2)(2)...(2)(2)(2))

(((1))) = ((a))

(((1)1)) = ((a+1))

(((1)(1)) = (((1)a))

(((2))) = (((1)(1)(1)...(1)(1)(1))) w/ a (1)'s

((((1)))) = (((a)))

(((((1))))) = ((((a))))

Sub-extensions

My Extension

This is my extension:

(1)2 = (((...(((a)))...))) w/ a levels

(2)2 = Arrow((2)2)

((1))= (a)2

((1)1)2 = Arrow(((1))2,((1))2,((1))2)

((1)(1))2= ((1)a)2

((2))2 = ((1)(1)(1)...(1)(1)(1))w/ a (1)'s

...

((1)2)2 = (((...(((1)))...)))w/ a levels

(1)= (((...(((1)2)2)2)...)2)2)w/ a levels

(n)(1)= (n)

(n)(1)1 = (n)a+1

(n)(1)(1) = (n)(1)a

(n)(2) = (n)(1)(1)(1)...(1)(1)(1) w/ a nestings

...

(n)(m)(1) = (n)(m)a

...

(1)1,2 = (a)(a)(a)(a)...(a)(a)(a) w/ a levels

(1)2,2 = ((((...(((1)1,2)1,2)1,2)...)1,2)1,2)1,2 w/ a levels

(1)(1),2 = (1)a,2

...

(1)(1)1,2 = (1)(1)(1)(1)(1)...(1)(1)(1) w/ a levels

(1)1,3 = (1)(1)(1)...(1)(1)(1)1,2  w/ a levels

...

Googleaarex Extension

(1,2) = ((...((a))...))/w a ()'s inside out.

(1,3) = ((...((a,2),2)...,2),2)/w a ()'s inside out.

(1,1,2) = (1,(1,...(1,(1,a))...))/w a ()'s inside out.

(1,1,1,2) = (1,1,(1,1,...(1,1,(1,1,a))...))/w a ()'s inside out.

...

(1[2]2) = (1,1,1,...,1,1,2) w/ a entries

(1,2[2]2) = (((...(((1[2]2)[2]2)[2]2)...)[2]2)[2]2)[2]2) w/ a nestings

(1[2]3) = (1,1,1,...,1,1,2[2]2) w/ a entries

(1[2]1,2) = (1[2](1[2](1[2](...(1[2](1[2](1[2]2)))...))) w/ a nestings

(1[2]1[2]2) = (1[2]1,1,1,...,1,1,2) w/ a entries

(1[3]2) = (1[2]1[2]1[2]...1[2]1[2]1[2]2) w/ a rows

...

(1[1,2]2) = (1[(1[...(1[(1[a]2)]2)...]2)]2)/w a []'s inside out

(1[1,3]2) = (1[(1[...(1[(1[a,2]2),2]2)...,2]2),2]2)/w a []'s inside out

(1[1,1,2]2) = (1[1,(1[1,...(1[1,(1[1,a]2)]2)...]2)]2)/w a []'s inside out

...

(1,22) = (1[1[...[1[1[a]2]2]...]2]2) w/ a nestings

(2,22) = Arrow((1,22))

(1,2,22) = (((...(((1,22),22),22)...),22),22),22) w/ a nestings

(1,23) = (1[1[...[1[1[a]2]2]...]2]2,22)

(1,21,22) = (1,21[1[...[1[1[a]2]2]...]2]2) w/ a nestings

...

(1[2]22) = (1,21,21,2...,1,21,21,2) w/ a second level entries

(1[3]22) = (1[2]21[2]21[2]2...1[2]21[2]21[2]2) w/ a second level rows

...

(1,32) = (1[1[...[1[1[a]2]2]...]2]22)

...

SuperJedi224/Googleaarex Extension

Let f(x) equal ((...(x)...)) with x sets of parentheses.

Let g(x) equal f(f(...f(x)....)) with x "f"'s

f() = (1)(), g() = (2)(), etc.

...

Let define f(x)(y) = (((...(((x)))...)))(y) w/ x levels

Let define g(x)(y) = f(f(f(...(f(f(f(x))))...))(y) w/ x levels

Let define f(x)(y) = (1)(x)(y), g(x)(y) = (2)(x)(y), etc.

...

f(a)(b)(c) = (((...(((a)))...)))(b)(c) /w a ()'s nested out

f(a)(b)(c)(d) = (((...(((a)))...)))(b)(c)(d) /w a ()'s nested out

...

(n)(m)l = (n)(m)(m)(m)...(m)(m)(m) w/ l entries

(n)(m)(1) = (n)(m)a

...

F Function

F_0(n) = Arrow((100,100100)(100)(100)...(100)(100))/w n (100)'s

F_x+1(n) = F_x(F_x(...F_x(F_x(n))...))/w n F's

F_a(n) = F_a[n](n)

F_0,1(0) = F_100(100)

F_0,1(n) = F_(F_0,1(n-1))(100)

F_1,1(0) = F_0,1(100)

F_1,1(n) = F_0,1(F_1,1(n-1))

F_0,2(0) = F_0,100(100)

F_0,2(n) = F_0,(F_0,2(n-1))(100)

F_1,2(0) = F_0,2(100)

F_1,2(n) = F_0,2(F_1,2(n-1))

...

...

...

If there is something i did not put in, just tell me.

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