12/06/2011, 08:12 PM
Ah, OK. Now I think I understand what you're talking about. There is a lot of terminology for this kind of thing, and I think it would be good to review it. In the case of addition, it's called N-ary summation, and in the case of multiplication, it's called N-ary products. In computer science, there are also names for NOPT, such as: apply, fold, map, reduce, zip, and many other terms. The most notable have special terms for left-associative nesting and right-associative nesting: foldl and foldr respectively. However, there are many other possibilities, some of which Henryk (BO) discussed in his Ph.D. thesis:
http://eretrandre.org/rb/files/Trappmann2007_81.pdf
Part of the issue in relating NEPT and hyperoperations is that you have to fix the hyperexponent so you can compare a^^6 and a^(a^a)^(a^a^a) for example. Even then, I don't think that any arithmetic shortcuts will come from this, I think the best that we can hope for in this case will be order relations and inequalities.
Regards,
Andrew Robbins
http://eretrandre.org/rb/files/Trappmann2007_81.pdf
Part of the issue in relating NEPT and hyperoperations is that you have to fix the hyperexponent so you can compare a^^6 and a^(a^a)^(a^a^a) for example. Even then, I don't think that any arithmetic shortcuts will come from this, I think the best that we can hope for in this case will be order relations and inequalities.
Regards,
Andrew Robbins