Thematics Of Order

[Jon Awbrey]

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TOO.  Note 7

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Re: TOO 6.  http://stderr.org/pipermail/inquiry/2005-April/002553.html
In: TOO.    http://stderr.org/pipermail/inquiry/2005-April/thread.html#2541

My memory server reminds me, however tardily,
that I don't really want to classify ordered
k-tuples with repeated elemnts as negligible,
in the way that I suggested last time, but I
will continue to put them aside for the time
being, until we get a better sense of how we
want symmetry groups to act on k-tuples with
all elements distinct.

Returning to the example we had last time, one arrangement
that suggests itself is to place the permuted 3-tuples and
their order measures on a permutation lattice of this form:

o-----------------------------------------------------------o
|                                                           |
|                      1 2 3 ~~> 2250                       |
|                             o                             |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                   (1 2) /   |   \ (2 3)                   |
|                        /    |    \                        |
|                       /     |     \                       |
|                      /      |      \                      |
|     2 1 3 ~~> 1500  o       |       o  1 3 2 ~~> 1350     |
|                     | \     |     / |                     |
|                     |   \   |   /   |                     |
|                     |     \ | /     |                     |
|               (2 3) |     (1 3)     | (1 2)               |
|                     |     / | \     |                     |
|                     |   /   |   \   |                     |
|                     | /     |     \ |                     |
|     3 1 2 ~~>  600  o       |       o  2 3 1 ~~>  540     |
|                      \      |      /                      |
|                       \     |     /                       |
|                        \    |    /                        |
|                   (1 2) \   |   / (2 3)                   |
|                          \  |  /                          |
|                           \ | /                           |
|                            \|/                            |
|                             o                             |
|                      3 2 1 ~~>  360                       |
|                                                           |
o-----------------------------------------------------------o

In this Figure, the labels on the edges of the lattice indicate
the pair of elements, among those known to be ordered 1 < 2 < 3,
which are transposed in passing between the two permutations of
3-tuples indicated on the adjacent nodes of the lattice.  It is
clear that this elevates the 3-tuple that is the most in order,
namely <1, 2, 3>, to the top of the lattice, while the 3-tuple
that is the least in order, namely <3, 2, 1>, settles to the
bottom of the lattice.  Moreover, the total ordering of the
natural number measures (or goedel numbers) is partially
reflected in the partial ordering of this lattice.

Thus we can say, on purely multiplicative grounds,
that we know the following chains of inequalities:

   360 < 540 < 1350 < 2250

   360 < 540 < 1500 < 2250

   360 < 600 < 1350 < 2250

   360 < 600 < 1500 < 2250

But for all we know on present evidence
the rest of the intermediate elements
remain incomparable among themselves.

Jon Awbrey

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