Thematics Of Order -- Discussion

[Jon Awbrey]

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TOO.  Discussion Note 3

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

Re: TOO 2.  http://stderr.org/pipermail/inquiry/2005-April/002545.html
In: 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: Inspired by this I propose a more practical question:
    Can we find a nice "closed" (or somehow easily computed)
    form for:

AK: http://www.research.att.com/projects/OEIS?Anum=A061858

AK: or a similar table involving the differences of ordinary
    integer multiplication table and e.g. the mult. table of
    GF(3)[X].

Antti,

That sounds like an interesting line of inquiry --
one of the reasons why I'm excavating these old
questions in a personal developmental manner is
that one person's huis clos is often another's
autobahn -- but I am struggling to maintain my
focus on the door I came in by, so I will have
to keep doing that for a while.

I do see the connection you are making with logic, though --
back when we used to call these "mask operations", it was
said that ADD was a combination of XOR plus a shifted AND,
and of course the process would have to be iterated until
it terminated.

For example:

0101 = x
0011 = y
----
0110 = x_1 = XOR(x, y)
0010 = y_1 = AND(x, y)*2
----
0100 = x_2 = XOR(x_1, y_1)
0100 = y_2 = AND(x_1, y_1)*2
----
0000 = x_3 = XOR(x_2, y_2)
1000 = y_3 = AND(x_2, y_2)*2
----
1000 = ADD(x, y)

Let me observe, however, that you are still staying largely within
the arena of additive representations, since base representations
are sums of products, or sums of multiples of powers of the base.
Here the main connective is additive, so maybe this would be
roughly analogous to disjunctive normal forms in logic.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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