Thematics Of Order -- Discussion

[Jon Awbrey]

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TOO.  Discussion Note 1

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AK = Antti Karttunen
JA = Jon Awbrey
PC = Peter Cameron

Re: TOO.  http://stderr.org/pipermail/inquiry/2005-April/thread.html#2541

In part:

JA: How much of the additive, linear, or total order in the
    sequence of natural numbers is "purely multiplicative",
    that is, how much of the order can be determined
    solely by inspection of primes factorizations?

PC comments:

PC: I think it was Ulam who remarked that there are
    rings which are multiplicatively isomorphic to
    the integers but additively completely different.
    One striking example is the polynomial ring over
    the integers mod 3.

AK comments:

AK: Please give us an example, how do you induce/produce permutations of
    the Natural numbers this way, or tell us whether there are already 
    illustrative specimens in the OEIS.  (It is then much easier for
    my concrete mind to grasp the concept ...)

Antti, Peter,

I'm reporting on some diversions that have diverted me for many years,
and this whole arena is littered with a host of tantalizing analogies
and disappointing disanalogies.  Memory is mercifully dim about some
of the avenues and deadends, but I can remember as a kid looking at
what I used to call the "vector logarithm" of a natural number n,
its sequence of exponents <e_i> on the sequence of primes <p_i>
and wondering if there were "structure constants" <a_ijk> that
would make addition into an algebraic operation on the lists
of primes factorization exponents.  Much diversion was had.

But for the sake of this excursion I'm really just about to
throw away almost all of the properly algebraic information
and ask a mutated version of the earlier question, to wit:

How much of the additive, linear, or total order in the
sequence of natural numbers is "purely combinatorial",
that is, how much of the order can be determined
solely by inspection of the "riffs" or "rotes"
of natural numbers?  Definitions to follow,
but in the meantime:

Cf: http://www.research.att.com/projects/OEIS?Anum=A061396
Cf: http://www.research.att.com/projects/OEIS?Anum=A062504
Cf: http://www.research.att.com/projects/OEIS?Anum=A062537
Cf: http://www.research.att.com/projects/OEIS?Anum=A062860
Cf: http://www.research.att.com/~njas/sequences/a061396a.txt

Jon Awbrey

PS.  Because of its cross-disciplinary nature,
     this discussion is being cross-posted to
     several different lists -- anybody who
     doesn't what their comments blogged
     please let me know.  Thanks, JA.

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