Riffs & Rotes & A061396

[Jon Awbrey]

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

Marc,

I, like you, am forced to chase my yarn through this maze --
not wooly of my our own making, let us hope -- and I fear
the order of disorder that looms each time that I dare to
jump around in time, or even to fray from the skein a bit.
So let me make amends in advance if I fail to pursue your
questions far enough to answer them as fully as I might.
But I cannot resist interweaving one of two bits below.

Marc LeBrun wrote:
> 
>  > Jon Awbrey
>  > This whole genus of questions is what I mean
>  > by:  Is there any "purely combinatorial" (PC)
>  > reason for the usual ordering of the natural
>  > numbers, one that bases itself wholly on the
>  > indicated indices of multiplicative structure?
> 
> I certainly haven't been able to absorb as much
> of this interesting material as I'd like, but
> if I've followed this much correctly I think
> the answer to this question must be "no".

Oh, sure.  I am deliberately leaving the meaning of "purely combinatorial"
a little bit vague, as I cannot imagine a definition that everybody would
agree on.  Just think of this as an intellectual exercise that starts out
with a bag of axioms, to which we can add or subtract this or that notion
of the "purely combinatorial" as the fancy may strike us at given moments.
The spirit of this game is just to look at the riff structures as objects
existing in their own right, and to ask what brands of orderings might be
natural to their species.  There are some obvious candidates for ordering
principles, and then there are rules that are certain to be controversial.
Having bracketed away what we know of arithmetic, divisory and successory,
for a while, we return now to the scene of natural numbers per se, asking
how much of their natural order we have, by our maxim, been able to mimic.

> The multiplicative structures are independent
> of the actual values of the primes.  If you
> were to swap 2 and 3, say, certain trees
> would just map onto different integers.
> But the set of all trees would still
> cover all the integers.

Well, this is the realm of "abstract primes" or "generalized primes" --
I have forgotten the proper name -- I do remember reading bunches of
stuff from Knopfmacher or Rademacher about it -- I now forget which,
maybe it was both -- anyway, so I think we are in the same ballpark.

But in the riff way of looking at natural numbers, there is something
else going on besides just the multiplicative structure, which is why
I took aim to say "the indicated indices of multiplicative structure".
To wit, there is that n-th prime function being dragged into the fray.
That rends asunder the erstwhile symmetry between 2 and 3 and parcels
different quanta of pendulosity out to them, namely 2 = p and 3 = p_p.

> Moreover, the primes involved might just as well be those of,
> say, the Gaussian integers or other such extension.  Then,
> of course, the idea of ordering itself becomes tricky.

It's already tricky enough for me,
but, of course, it is well spoken:

"Many such journeys are possible."

And you know the canonical reply.

Many Regards,

Jon Awbrey

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤