Recently I've been kind of curious about tetration, specifically why it doesn't introduce any new inverse functions in the way lower operations do- addition needs subtraction, multiplication needs division and exponentiation needs roots and logs (two inverses because it isn't commutative.) So if the pattern continues, if the next diagonalization of the successor function doesn't perfectly inherit the properties of its predecessor then it should introduce a new inverse. But every description of tetration I see says that it inherits the log and root from exponentiation as the superlog and superroot, but exponentiation loses a fundamental property when it's iterated: associativity. $x^{x^x}$ is different from ${x^x}^x$.
My first theory was that we take for granted that tetration refers to "top-down" associativity because "bottom-up" is essentially just multiplicative exponentiation, ${(x^x)^x = x^{2x}}$ but it could just as easily be said that roots are just a change in associativity for formulas like ${{(x^y)}/z} \ne {x^{(y/z)}}$ That operation created gaps in the number line, but that led us to the complex numbers. So that made me think, could we have a constructive extension of the number line created by tetration, and could that reconcile the gaps in our ability to define non-integer power tower heights with a single analytic function?
$\newcommand{\vc}[3]{\overset{#2}{\underset{#3}{#1}}}$
Unfortunately this theory didn't go anywhere; if I defined a value where ${f({x^{x^x}}) = {{x^x}^x}}$, it would still have an infinite number of solutions in the complex plane.
The reason imaginary numbers are useful because they can be uniquely defined in terms of existing numbers; even though 'i' doesn't have a numeric solution, it provides a unique and reversible solution to the equation ${x = (a+bi)^y}$ for any real numbers a,b and y.
With ${z = {x \uparrow\uparrow y}}$, we don't have a unique solution for y given only x or vice versa. However, there are countably infinitely many roots which can be represented as a generalized tetrated form of the Lambert W function. (https://math.eretrandre.org/tetrationforum/attachment.php?aid=1215) because ${x^{x^{x^...}} = e^{ln(x)*e^{ln(x)*e^{...}}} = e^{W(ln(a + 2\pi bi))^{W(ln(a + 2\pi bi))^...}}}$ where ${a + 2\pi bi}$ = x.
Using this property, we can reduce to a single solution with two parameters: the branch of the W function and the complex component of x reduced by a factor of ${2\pi}$. So theoretically, an analytic description of tetration wouldn't have 4 inverse relationships, but 5.
Of course we generally take for granted that we use the -1 and 0 branches for the W lambert function, but with 5 variables we can define a unique relationship with tetration. To give each part of the relationship a name:
$$x = Base$$ $$y = Height$$ $$z = Tetration$$ $$b = Branch$$ $$t = Twist$$
Given this complexity, let's change the notation: $$x = \vc{\overline{\sqrt[y]{z}}}{bWt}{}$$ $$y = \vc{slog_{x}{z}}{bWt}{}$$ $$z = x \vc{\nearrow}{}{bWt} \vc{y}{}{}$$ $$b = x \vc{\nearrow}{}{Wt} \vc{y}{z}{}$$ $$t = x \vc{\nearrow}{}{bW} \vc{y}{z}{}$$
There isn't an analytic function for non-integer values for y, b or t, but a cross-sectional one can be generated provided you have all but one of these variables defined.
Let's say you have $$256 = 2 \vc{\nearrow}{}{0W(x)} \vc{3}{}{}$$
This means you have a power tower of three 2's equal to 256 whose solution is on the 0 branch of the W function, which is fine when t is an integer, but when t is .5, x must have the form ${a + (.5)* 2\pi i}$ for a real value a, meaning it must have an imaginary component that is defined independently of i, which 2 does not. So to reconcile that, we would need a hypercomplex value.
If you define specific values for these numbers, theoretically you could map out a 4d topological map of tetration, filling in the holes between the branches of the W function and heights for their respective power towers.
I realize this isn't a super rigorous proof, it was just an idea I've been playing around with, so the main things I'm wondering is if:
- A: This is all actually a constructive solution or if I've been making some leaps on logic, particularly with the lambert W function.
- B: If it is valid, would 5 be the minimum number of variables or could it be reduced further?
- C: What's the minimum algebra that would be necessary to describe these values- Quaternions, a Bicomplex or something else?
- D: Could the same reasoning be extended to pentation, hexation, etc.?