Notations and Opinions

[andydude]

...
y = x{4,b}h for iterated exponentiation beginning at x: b^...^b^b^x
(and from earlier discussions)
y = x{3,b}h for iterated multiplication beginning at x: x*b^h
y = x{2,b}h for iterated addition beginning at x: x+b*h

for my needs for the time being,
the "height"-function
h = hgh(x,b)
if
x = 1 {4,b} h
= b [4] h // related to the tetrational notation
...

I personally think that the Arrow-Iteration-Section notations discussed in my first post cover most of these use cases, but U-tetration is different enough to require a special notation. Here are my recommendations:

\(
\begin{tabular}{l|c|c}
\text{ASCII} & \text{Arrow-Iteration-Section} & \text{Special} \\
\hline
\mathtt{x\^\^y(a)}\text{ or }\mathtt{(x\^)\^y(a)}
& (x {\uparrow})^y(a)
& \exp^y_x(a)
\\
\mathtt{x\^\^{\backslash}z(a)}
& (y \mapsto (x {\uparrow})^y(a))^{-1}(z)
& \text{slog}_x(z) - \text{slog}_x(a)
\\
\mathtt{z/\^\^y(a)}
& (x \mapsto (x {\uparrow})^y(a))^{-1}(z)
& \
\\
\hline
\mathtt{x\^-\^y(a)}\text{ or }\mathtt{(x\^-)\^y(a)}
& (t \mapsto x^t - 1)^y(a)
& DE^y_x(a)
\\
\mathtt{x\^-\^{\backslash}z(a)}
& (y \mapsto (t \mapsto x^t - 1)^y(a))^{-1}(z)
& \
\\
\mathtt{z/\^-\^y(a)}
& (x \mapsto (t \mapsto x^t - 1)^y(a))^{-1}(z)
& \
\\
\end{tabular}
\)

but I've seen other notations elsewhere. The one I've seen used the most is x^^y@a, although I had also used y`x`a in the past. Also, GFR uses x$y*a or something like that, which I find confusing. Thats all about iter-exp.

Starting from scratch using Arrow-Iteration-Section notation, we find that the natural expression in ASCII is (x^)^y(a) which could be shortened to x^^y(a) which means the corresponding notation for iterated decremented exponentials is (x^-)^y(a) which could be shortened to x^-^y(a), what do you think? About iter-dec-exp/U-tetration, this would mean that your "height" function is h = hgh(x, b, a) = b^-^\x(a) and h = hgh(x, b) = b^-^\x which I would've called the "super-decremented-logarithm" or something.

We might even go so far as to use similar notations for superroot and superlog, so srt_n = (/^^n) and slog_b = (b^^\).

While I'm at it, I might as well summarize the other suggestions (based on BO's):

\(
\begin{tabular}{l|c|c}
\text{ASCII} & \text{Arrow-Iteration-Section} & \text{Special} \\
\hline
\mathtt{x\^\^y}
& x {\uparrow}{\uparrow} y
& {}^{y}{x}
\\
\mathtt{x\^\^{\backslash}z}
& (x {\uparrow}{\uparrow})^{-1}(z)
& \text{slog}_x(z)
\\
\mathtt{z/\^\^y}
& ({\uparrow}{\uparrow} y)^{-1}(z)
& \
\\
\hline
\mathtt{x[n]y}
& x {\uparrow}^{n-2} y
& x \begin{tabular}{|c|}\hline n \\\hline\end{tabular} y
\\
\mathtt{x[n]{\backslash}z}
& (x {\uparrow}^{n-2})^{-1}(z)
& {}^{n}_{x}\begin{tabular}{|c}z \\\hline\end{tabular}
\\
\mathtt{z/[n]y}
& ({\uparrow}^{n-2} y)^{-1}(z)
& {}_{n}^{y}\begin{tabular}{|c}\hline z \\\end{tabular}
\\
\hline
\mathtt{x[n]\^y(a)}
& (x {\uparrow}^{n-2})^y(a)
& \
\\
\mathtt{x[n]\^{\backslash}z(a)}
& (y \mapsto (x {\uparrow}^{n-2})^y(a))^{-1}(z)
& \
\\
\mathtt{z/[n]\^y(a)}
& (x \mapsto (x {\uparrow}^{n-2})^y(a))^{-1}(z)
& \
\end{tabular}
\)

I must say, the slash notation is by far the most expressive tetration notation I've ever seen. It allows full expression of practically anything I can think of that is hyperop/tetration related. As you can see, it covers many topics that do not have a specialized notation yet.

Andrew Robbins