01/22/2008, 07:04 PM

Ivars Wrote:Another question is to the whole concept of starting iteration from the end, from right as opposed to usual. I think that is the most important thing that distiquishes tetration from other operations, and , if analoques to such approach could be constructed in any other operation of function, it would be very much of interest to study them.

So that notion, iterating from right ( in my terminology from end - or perphaps it is actualy the beginning?) is very important to clarify to an extent that we can say e.g.

What you are talking about only applies to binary operations. Integer iteration of one-variable functions is always unique, no matter what. However, when discussing binary operations (which are two-variable functions), there is more than one way to construct a one-variable function from a two-variable function. This is where all the possibilities come from, from binary operations, not from iteration. The iteration of a binary operation could be described as right-iteration, and left-iteration, and each of these is pretty unique. With right-iteration of the binary operation B we get (xB(xB(xB...B(xBy)))), which actually has 3 parameters (x, y, and n -- the iterator). With left-iteration of the binary operation B we get ((((xBy)B...By)By)By) which also has 3 parameters (x, y, and n).

What the Bromer-Mueller arrow allows (which Henryk mentions also) is that you can alternate between left-iteration and right-iteration however you want, and this is where you get things like iterated powers (left-iterated exponentiation or lower-tetration), and iterated iterated powers (left-iterated left-iterated exponentiation, or lower-pentation). This gives a binary tree of hyper-operators from exponentiation, which I call mixed hyper-operators. Each of the hyper-operators in this binary tree can be described by a rank (like GFR uses), but instead of being unique by rank, there is 1 rank-2 operator, 1 rank-3 operator, 2 rank-4 operators (tetration and lower-tetration), 4 rank-5 operators, 8 rank-6 operators, 16 rank-7 operators, and so on. Each time you have a choice of left-iteration or right-iteration, so the number of operators doubles. The terminology I prefer uses pure iteration rather than right/left-iteration, which means right-iterated exponentiation = iterated exponentials, and left-iterated exponentiation = iterated powers.

Now iteration aside, there are other ways to associate binary operations. If you consider expressions rather than left-iteration and right-iteration of a binary operation, then there is no implied associativity. With no implied associativity, you could have expressions like (aB((bB(cBd))Be)) or something like that. A special case of these when all the elements are the same is what Henryk considers, I think. What this reduces to is a set of hyper-operators on binary trees, which is mind-bending and very hard to cope with. These operators are much more general than mixed hyper-operators, and they should not be confused at all. One thing I noticed that I included in the huge-FAQ is that Henryk calls this "repeated exponentiation" which I believe is very appropriate.

Andrew Robbins