This is a backward (and possibly frontward) extension to the Fast-growing Hierarchy (FGH)

Each Unique Semicolon-Particle (a group of letters defining a notation, hierarchy, operation also comes before a semicolon) can describe an operation, a notation, or even an hierarchy


--Regular Fast-growing Hierarchy (FGH) (also, R = Recursion)

F(R0;n) = n+1

F(Ra+1;n) = F^n_a(n) where f^n is Function Iteration

FRa;n) = F^n_a[n](n) if a is a limit ordinal

But: when a is only an integer

--Operators and linear BEAF arrays

F(aA;n,m,x...) = a+n+m+x... {S for subtraction, [(a-n)-(m-x...)] = [((((...(a-n)-m)-x)-...))]}

F(aM;n,m,x...) = a*n*m*x*... {D for division, [(a/n)/(m/x...)] = [((((...(a/n)/m)/x)/...))]}

F(aP;n,m,x...)= a^n^m^x^... (Power Towers; R for Roots, aR(n,m) = a^(m/n) )

F(aUm;n) = a{m}n = {a, n, m}

F(aUm,n; x, y, z ...) = {a, n, m, x, y, z...}

--Named but unquanitied (not given a numeric value, like Rayo's number) numbers

F(<<first,second...> (important) letters>;n) (if name has less than or equal to 3 letters write it in full; ex: Rayo -> Ryo and Sam -> Sam)

Ex: Fish Number 5 = F(FshNmbr;5), Rayo's Number = F(_RyoNmbr;1)


F(aF;n)= a!n 

F(aHF;n, m, x, y, z...) = a![n, m, x, y, z...] (when n=1, then aHF -> aF and (1, #) -> (1))

F(aMF;n) = a!!<n>!! (when n=1, then aMF -> aF)

F(aFF;n) = (a)!_n (falling factorial)

F(aRF;n) = a^(n) (rising factorial)

F(aAF;n) = (alternating factorial of a+n)

--Other Hierarchies F_Ln(n) = Faster Growing Hierarchy

F(S;n) = Slow-growing Hierarchy

F(H;n) = Hardy Hierachy

--Extensible E system(s) (If a=10, then Ea -> E)

F(Ea;n, r1, r2, r3...) = E[a]n#r1#r2#r3...

F(Ea;nhb,chd,ehf...) = E[a]n#c#<d>#e#<f>#...

compounds and cascaders: #=h,^=p,*=m

so: a(#^#*#^#)b = a(hphshph)b

Table Summarizing Semicolon-Particles (SP):



A, S, M, D +, -, *, /
E Extensible E System (S.S)
U Knuth Up-Arrows, BEAF (J.B)
F, HF, MF Factorial, Hyperfactorial (Aarex), Multifactorial
P Exponentation (Power Towers)
R,N Recursion/Nesting
xxxxxxxxxxxxxxxxxxxxx <Suggestion here>
xxxxxxxxxxxxxxxxxxxxx <Suggestion here>

Compounds and Other Functional Groups

F(O1:O2#aX/bY/...) = O2(O1(aX,bY,...)) A+,S-,M*,D/,R(epeats) (123R4 = 123,123,123,123 [as a single number])

Exception(s): are functions in f(x) form such as sin, cos, Log, exp, erf, etc.

Generally: F(X:Y:Z...) = X(Y(Z)))

Simplified Math Relation Line Notation00:52

Simplified Math Relation Line Notation

This may be outdated just replace the semicolons with commas and the colon with a hash

F(PR6#E2/E[23]4) = (10^2^23^4)^(10^2^23^4)^... 6 times

F(13R9) = 13 repeated 9 times  =  131,313,131,313,131,313


Non-Googological Concepts!

Other structures can be defined, such as Polynomials:

F(Pn;a,b,c...f) = a*x^n + b*x^(n-1) + c*x^(n-2) + ... + f

F(P2; 4,2,0S13) = 4x^2 + 2x - 13 ; x^3+ x - 2 = F(P3;1,0,1,0S2) ; 3*x^4 - 9 = F(P4;3,iii,9) ; 

And Functions: sin(x) = F(x\sin) and generally G(x,a,b...) = F(x\G,a,b,...) [a,b,...k are secondary, tertiary, and n-ary numbers of function G)

Defined Number Groups:

Number groups depictable
Group Example
Rationals F(3D2) = 3/2; 2.34 = F(117D100)
Algebraic Irrationals

sqrt(5) = F(P; 5/1D2) or F(5R1t2)

17th root of 81^5 = F(81R5t17)

Transcendental Irrationals

(constants are the symbol)

sin^3(32) = F(32/sin,3)

pi = F(pi); e = F(e); i = F(i) or F(0S1R1t2)

Complex/Imaginary 3+2i = F(A:3/2Mi); 7i = F(7Mi)
Integers -n = F(0Sn) [n>0]; 0 = F() or F(0)


Hypercomplex (quaternions, Sedenions, Octonions...)

{a,b...|...d,c} = F(Sr:a,b...//c,d...)

a*e_0 + b*e_1 +c*e_2... = F(Hc:a,b,c...)


Basically an Expression is simply a giant function: [also a^(m/n) = F(aRmtn); if m=1 then; a^(1/n) =F(aRtn)]

sqrt(2) - 7 = F(S:2R1t2/7) [a : signifies the end of that operation, if its at the end of everthing it can be omitted]

(Log_3(7-3*sin(4)) +9*sqrt(5))/8 = F(D:A:S:7/3M4\sin//Log3///9:/9M5Rt2:8)


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(more to possibly come!)

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