# Tetration Forum

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Ivars Wrote:........................................
Sometimes I feel that with pentation .......................................
I can not see math as not being possible to close logically while allowing it to develop because of undetermined symbols like infinity, etc..

We are still working to clarify what happens up to the s=4 hyper-operation rank, i.e. at the tetration level. In particular, concerning the "priority to the right" business, we have:

(s=1, addition): a+(a+(a+a)) = a x 4 [bracketing not necessary, add. commutable, associable]
(s=2, multipl.): a.(a.(a.a)) = a ^ 4 [bracketing not necessary, mult. commutable, associable]
(s=3, expon.): a^(a^(a^a)) = a # 4 [bracketing indispensable, exp. non-commutable, non-assoc.]
(s=4, tetrat.): a#(a#(a#a)) = a § 4 [bracketing indispensable, tetr. non-commutable, non assoc.]

Symbol § is provisionally used here for "pentation". However, concerning the same priority business, for s=5 we must also have:

(s=5, pent.): a§(a§(a§a)) = a ç 4 [bracketing indispensable, tetr. non-commutable and non-associable]. Symbol ç means here "hexation". And ... so on.

The problem is that we have not fully analyzed yet the fourth rank level. We need to show that "sexp", "slog" and sroot" are smooth uniquely defined (1- or 2-valued) "functions", before proceeding further. Therefore, we are not ready to "sublime" up to the fifth rank.

Objects like "undeterminations", "infinitesimals" and "infinities" may strongly intervene in the analysis process, but I don't see them as really serius obstacles, if we pay the necessary attention, at this stage. I agree that the study of the hyper-operations, starting from the tetration rank might give new ideas also for a more complete approach to undeterminate and infinite magnitudes. We shall see!

GFR
THIS POST WAS ORIGINALLY POSTED IN THE ZERATION THREAD BECAUSE THERE ARE IDEAS THAT RELATE TO ZERATION THAT I HAVE POSTED THERE. I THOUGHT I WOULD POST THIS HERE BECAUSE IT RELATES MORE TO NOTATION AND OPINION. ENJOY!
///The notation won't display right because the original post was deleted.. give me some time to fix this.... sorry!

Here is the notation I find easy to work with in my notebook. Tell me what you think.

Regular Operations

x[attachment=268]y = z

Left Inverse (Stick on the Left)

x = z[attachment=267]y

Right Inverse (stick to the right)

y = z[attachment=270]x

The Minus One Law in My Notation
(x[attachment=269]b) [attachment=270] (x) = x [attachment=269](b-1)

Logarithmic Notation
Also I am in the midst of developing a fractional notation as well as horizontal notation for logarithms.
log x (base 10) looks like this

x
~
10

or x~10
where the ~ is like a fraction line. I find the current/traditional notation clumsy.

for example
Code:
((a~b)~c) = log (log  a)                c     b and a~(b~c) = log     a              log b                 c
Anyway, I find the combination of the fractional and horizontal easier to manipulate.

Enjoy!
James
What we really need is not a new notation, but a notation that can be/is widely agreed on and that widely can be used.
Basicly, as the overwhelming majority of mathematical articles is written in (La)TeX (which is also an input option on this forum), we need an ASCII notation and a TeX notation.

A very good example is the by Gianfranco (not even officially) introduced ASCII notation [n] for the nth hyperoperation. Its very intuitive, good readable, and everyone is immediately going to accept it.
However there are not yet that convincing proposals for the log-type and root-type inverse functions.
I think we commonly agree on the use of slog_b for the inverse of x->b[4]x. And for ssqrt as the inverse of x->x[4]2.

One generalization would be lg[n]_b for the inverse of x->b[n]x and rt[n]^k for the inverse of x->x[n]k.

From an algebraic standpoint however our considerations fall into the category of a quasigroup. That is a set with an operation * that has unique left and right inverses. There the left and right inverses are written as \ and / respectively. This is very mnemonic as one cancels always on the corresponding side:
(a*x)/x=a and (a/x)*x=a
x$$x*a)=a and x*(x\a)=a applied to our problem we should choose b $n] y as the logarithm type inverse, i.e. b \[n] ( b[n]x )=x and y /[n] k as the root type inverse, i.e. (b [n] k) /[n] k =b. then is slog_b(x)=b\[4]x and ssqrt(x)=x/[4]2. This is even a bit mnemonic as \ reminds slightly of an l (for logarithm) and / reminds slightly of an r (for root). x/[3]k also reminds slightly of x^(1/k). bo198214 Wrote:A very good example is the by Gianfranco (not even officially) introduced ASCII notation [n] for the nth hyperoperation. Its very intuitive, good readable, and everyone is immediately going to accept it. Thank you, Henryk, for your kind appreciation and comments. As a matter of fact, the first idea of a new and, possibly, clear graphical notation was born during my cooperation with Konstantin Rubtsov, in http://forum.wolframscience.com/showthre...readid=579 http://forum.wolframscience.com/showthre...readid=956. As you see in the annexes of the two threads, we agreed on a Box Notation of direct and inverse hyperoperations (of both the log and root types) using, unfortunately, symbols not belonging to the ASCII set. We called it the "KAR-GFR Box Notation". The square-bracketing notation of the "direct operations" was used by me for the first time in this Forum for simplifying the writing, without loosing information. I should like to confirm it here officially, transforming this fact in a kind of official proposal. bo198214 Wrote:............... (a*x)/x=a and (a/x)*x=a x\(x*a)=a and x*(x\a)=a applied to our problem we should choose b \[n] y as the logarithm type inverse, i.e. b \[n] ( b[n]x )=x and y /[n] k as the root type inverse, i.e. (b [n] k) /[n] k =b. then is slog_b(x)=b\[4]x and ssqrt(x)=x/[4]2. As you know, the KAR-GFR Box Notation included two half-boxes for indicating the hyperroot (similar to a capital Gamma) and the hyperlog (similar to a capital L), both superscripted (or underscripted) by the hyper-operation rank and accompanied by the appropriate bases or exponents. In a simplifyed version of them, valid only for rank 4, I "unofficially" used, in this Forum, the following notations: [k/]srt(x) : the k-th super-root of x [b$slog(x) : the superlog, base b, of x Now, your proposal, which takes into consideration various other facts and proposals, brings to: ssqrt(x) = [2/]srt(x) = x/[4]2 slog_b(x) = [b\]slog(x) = b$4]x. ................................................................................... As a first reaction, I should like to introduce a slight modification in your interesting proposals (to be identified as ... the GFR-BO, or BO-GFR simplified ASCII notation): ssqrt(x) = [2/]srt(x) = x/[4]2 slog_b(x) = [b$slog(x) = b[4]\x. (modification) In this case, for a direct hyperop such as y = b [n] k, we shall have: b [n]\ y as the log-type inverse, i.e. ---> b [n] \ ( b [n] k ) = k and y /[n] k as the root-type inverse, i.e. --> (b [n] k) /[n] k = b. The remaining problem is that this simplified ASCII notation would give: b [1] k = b + k = y --> b = y /[1] k = y - k, k = b [1] \ y = y - b b [2] k = b * k = y --> b = y /[2] k = y / k, k = b [2] \ y = y / b b [3] k = b ^ k = y --> b = y /[3] k = k-srt y, k = b [3] \ y = b-slog y. In other words, for rank 3 and > 3, we are in contrast with the traditional prefixed notation of the inverse operations. ................................................................................ We could then try a more schematical approximated notatiuon, such as: b [1] k = b + k = y --> b = y /1| k = y - k, k = b |1\ y = y - b b [2] k = b * k = y --> b = y /2| k = y / k, k = b |2\ y = y / b b [3] k = b ^ k = y --> b = y /3| k = k-srt y, k = b |3\ y = b-slog y. The advantage of this schematical notation is that we could admit an upside-down mirror inversion of the operation symbols, in their inverse sequence, like (see the third line): b [3] k = b ^ k = y --> b = k \3| y = k-srt y, k = y /3| b = b-slog y. A compromise of the straight and mirror schematical notation, for always showing a "prefixed" inversing operator (acting on y at its right) could be: b [3] k = b ^ k = y -> b = k \3| y = k-srt y, k = b |3\ y = b-slog y. In general, for: y = b [n] k, we might have: b = k \n| y = y /n| k, the root-type left-inverse, and k = b |n\ y = y |n/ b, the log-type left-inverse. This is my "official" additional proposal, hoping not to have created more noise that it is absolutely indispensable. Please tell me what you (BO, and ... all of you) think of it. Gianfranco Quote:b [3] k = b ^ k = y --> b = y /[3] k = k-srt y, k = b [3] \ y = b-slog y. In other words, for rank 3 and > 3, we are in contrast with the traditional prefixed notation of the inverse operations. I dont see this contrast, we have: $y /[3] k=y^{1/k}$ and $b [3]\backslash y = \log_b y$ (And by this you can easily remember that / corresponds to the root type.) I mean the side is anyway arbitrary, you also have ${^n x}$ but you write x[4]n. GFR Wrote:The advantage of this schematical notation is that we could admit an upside-down mirror inversion of the operation symbols, in their inverse sequence, like (see the third line): ... In general, for: y = b [n] k, we might have: b = k \n| y = y /n| k, the root-type left-inverse, and k = b |n\ y = y |n/ b, the log-type left-inverse. But Gianfranco thats confusing! \ and / for the same operation depending on which side. I dont want to first think a minute what is meant by the current symbol! There is also no mnemonics attached. Neither is a both side variant really needed nor is it usual to have it. There is no opposite side variant for -, / or ^. So if you really desperately need the both-side variants then keep the same operation symbol! E.g. /n| and |n/ as root-type inversion, but I dont see usage for them. And you have to burden your memory with another rule to decide on which side is the base/exponent, i.e. on the side which is not |. However I see a bit a problem with /n|, as when you use it without spaces it can be confusing, for example |a/n|x| = | a /n| x | I placed some attention to this problem when I was deciding for /[n] because you can not misread the / as a division (because it is followed by an open bracket). This ruled out the other variant that I had in mind: /n/. However your idea to put [n]\ instead of \[n] is a better one as you can better memorize the rule b[n]k /[n] k = b and b [n]\ ( b[n] k ) = k as "The thing to be reduced is on the (reducing) operation side (i.e. the side with the / or \ attached)" PS: We can call this the BO-GFR simplified ASCII notation, however by such discussions there always comes up the image of a commitee designing conventions (by long and intense democratic discussions) which dont fit real needs of the using people. While really useful things are made without commitees! However I hope its not the case here. Of course, I was joking. I spent my entire life in committess and commissions and I am not interested in participating in new ones, until the complete stop of the engines. After that, we shall see! I meant only to find a provisional "code" for easy future reference, in this Forum. The Future is in the hands of the gods. GFR That is interesting, I didn't even realize I was using GFR notation the last time I used [n], so it must be easy! Also, bo198214, I am amazed at how simple, your inverse hyperop notation is, I had heard of quasigroups before, but it didn't occur to me that they represented hyperops, but they clearly do. When it comes to exact notation, however, I would agree with GFR, in that the root notation "y /[n] k" should be left alone, and is acceptable as is, but the log notation "b [n]\ y" is something I would perfer, because it indicates that "b [n]\" is the function because the "\" clearly separates the "function" part from the "argument" part, and "y" is usually considered the primary argument. I vote for these two for the ASCII notation of inverse hyperops. Andrew Robbins GFR Wrote:Please tell me what you (BO, and ... all of you) think of it. Gianfranco Dear Gianfranco (and all) - the proposals are nice. However, noting, that we find a route for tetrational notation now, I feel I should explicitely re-remark, that for iterated exponentiation and decremented iterated exponantiation, with which I'm involved, I always need three parameters, one additional for the top-(beginning)-exponent x for y= b^..^b^x, height h (= b's in number of h, or more generally: h being the iteration-count) and I should step away a bit and use different symbols. So I better announce (the ascii-notation) 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 just to prevent confusion between these two concepts and the reading of my notes in context of the tetration-discussion. Gottfried Gottfried Wrote:... 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 yxa in the past. Also, GFR uses xy*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
andydude Wrote: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

Yes, this looks nice - maybe because we are already a bit used to it.

However, the collection of slashes and symbols for iterated exponentiation (IE, T) and especially for decremented iterated exponentiation (DIE, U) still looks a bit obfuscating to me - again: perhaps this is a matter of usage and experience.

While resorting to "hgh()" as function giving the height of a powertower I should add, that another -even more- natural function occurs with the notation x {4,b} h=y, since x already has a height; "h" is here actually the height-difference of the towers x and y in terms of "bricks" or "stones"; so what in tetration is the superroot (and subequently the srt-function) may be here the stone- or brick-function "stn" or "brk", where the latter had even some resemblance to a notation of base using b... .

So until I'm getting used to some slash-notation for IE (T()) and DIE (U()) I should propose the "brk"-function indicating the value of the base, which is needed to get from x to y if h times iterated... (well, I've no use for it so far, but...)

Gottfried
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