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:...............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.

(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.

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.

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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