G function

:Not to be confused with the G(n) function used to define Graham's number.
A G function is a type of pseudo-function used by Jonathan Bowers in his Array Notation.Size 4 Arrays If a G function has base b, we say that:
:\(G(a) = b\underbrace{\uparrow\ldots\uparrow}_ab = \{b, b, a\}\), using Arrow Notation and BEAF, respectively.
When the letter G is written by itself in a mathematical context, it simply means b. Therefore, we can write G = b.
When a G function is iterated over itself, Bowers omits the parentheses. For example, G(G(G(4))) could be written GGG4. If G was in base 4, then this could also be written GGGG, since G = 4.
We can commit an abuse of notation by treating concatenated Gs as multiplication, and then extrapolating to exponentiation, tetration, etc. Thus GGGG could also be written as G4, which is G "multiplied" (iterated) by itself 4 times. An extended example would be GG = G tetrated to G, which is equal to GGGG in base 4. Since the rightmost occurrence of G in a multiplicative expression is converted to the base, it is equal to G444, which expresses both the function \(n \mapsto G^{4^{4^{4}}}(n)\) and the constant 4444. Namely, we have the strict equation GGGG = G444 of either functions or constants. Therefore it does not give a structure essentially different from the simple iteration of G.
Although this community believed that we can even extend "G arithmetic" to BEAF by saying "GG is the same as {G, G, 2}", it is unformalised due to the lack of the definition of the expansion.
== Sources ==
== See also ==
*Graham's number
Category:Functions
Category:Jonathan Bowers
Category:Notations