Zeration

[quickfur]

Wah, I forgot to welcome my dear friend (hopefully he agrees with this designation ) quickfur - or sporadically also known as slowpaw - on this forum!

Yes, I agree with this designation. I feel quite honored, in fact.

Anyway, with regards to the {n} hierarchy, you're right, a class of solutions is given by \( a\{n\}b=\exp^{\circ n-1}\left(\ln^{\circ n-1}(a)+\ln^{\circ n-1}(b)\right) \).

It does have some interesting attributes, although probably only in very specialized fields. I've had somebody tell me once that they actually contemplated the same operations and found it useful in computer graphics: this class of operations can be used for image compositing where the order of composition doesn't matter, and the pixel value must remain within a specified range.

Another interesting observation (at least, interesting to me) is that the converse construction, \( a\oslash b = e^{\ln\ a + \ln\ b} = ab \). That is to say, in stepping from addition to the diamond operator, if we "stepped in reverse", we get to multiplication. In other words, the diamond operator is in some vague sense the "same distance" from addition as addition is from multiplication. But then again, this is just because we're remapping the field structure (+,*) to (♦,+), so we're not really discovering any new things.