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RATD. Note 15
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Cleaving a while longer to this geometric vein, let us now
revisit the earlier pair of 3-adic relations that I showed,
namely, L_0 and L_1 c X = X_1 x X_2 x X_3 ~=~ B^3, forming
a 2-projectively indiscernible couplet of 3-adic relations,
in the upshot, a pair of 2-projectively irreducible 3-adic
relations. This time, though, let's shift our perspective
a bit, viewing that pair's entanglement on a mat like this:
o-------------------------------------------------o
| |
| 111 |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| 110 101 011 |
| |\ / \ /| |
| | \ / \ / | |
| | \ / \ / | |
| | \ / | |
| | / \ / \ | |
| | / \ / \ | |
| |/ \ / \| |
| 100 010 001 |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \|/ |
| 000 |
| |
o-------------------------------------------------o
Figure 3. Boolean 3-Cube, X_1 x X_2 x X_3 ~=~ B^3
In this Figure, the points of B^3 are notated as bit strings of length three.
And its encompassing genre, one views the construction of the k-dic relations
L c B^k as a matter of coloring the nodes of the boolean k-cube with choices
from a pair of colors that stipulate points "in" or "out" of the relation L.
In reviewing the relations L_0 and L_1, I will plot the coordinates
of the points that are in the relation and blot out the coordinates
of the points that are not in the relation in question. By way of
computing and illustrating the 2-adic projections of the relations,
I will translate the coordinates of the points in each relation to
the plane of the projection, and then "dot out" with a dot "." the
part of the coordinate that is out of place on the plane in view.
Figure 4 shows L_0 and its three 2-adic projections:
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| 110 101 011 |
| |\ / \ /| |
| | \ / \ / | |
| | \ / \ / | |
| | \ / | |
| | / \ / \ | |
| | / \ / \ | |
| |/ \ / \| |
| o o o |
| \ | / |
| 11. \ | / .11 |
| |\ \ | / /| |
| | \ x y z / | |
| | \ \ | / / | |
| | \ \ | / / | |
| | \ \|/ / | |
| | \ 000 / | |
| | \ / | |
| 10. 01. .10 .01 |
| \ | | / |
| \ | | / |
| \ | | / |
| x y 1.1 y z |
| \ | / \ | / |
| \ | / \ | / |
| \| / \ |/ |
| 00. / \ .00 |
| / \ |
| / \ |
| / \ |
| 1.0 0.1 |
| \ / |
| \ / |
| \ / |
| y z |
| \ / |
| \ / |
| \ / |
| 0.0 |
| |
| Relation L_0 |
| |
| L_0 = {<x, y, z> in B^3 : x + y + z = 0 mod 2} |
| |
| L_0 contains just the following four |
| triples of the form <x, y, z> in B^3: |
| |
| <0, 0, 0> |
| <0, 1, 1> |
| <1, 0, 1> |
| <1, 1, 0> |
| |
o-------------------------------------------------o
Figure 4. Relation L_0 and Its Plane Projections
In short, the points of L_0, on several projection,
cover all of the points of all three copies of B^2.
Figure 5 shows L_1 and its three 2-adic projections:
o-------------------------------------------------o
| |
| 111 |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / \ / | |
| | \ / \ / | |
| | \ / | |
| | / \ / \ | |
| | / \ / \ | |
| |/ \ / \| |
| 100 010 001 |
| \ | / |
| 11. \ | / .11 |
| |\ \ | / /| |
| | \ x y z / | |
| | \ \ | / / | |
| | \ \ | / / | |
| | \ \|/ / | |
| | \ o / | |
| | \ / | |
| 10. 01. .10 .01 |
| \ | | / |
| \ | | / |
| \ | | / |
| x y 1.1 y z |
| \ | / \ | / |
| \ | / \ | / |
| \| / \ |/ |
| 00. / \ .00 |
| / \ |
| / \ |
| / \ |
| 1.0 0.1 |
| \ / |
| \ / |
| \ / |
| y z |
| \ / |
| \ / |
| \ / |
| 0.0 |
| |
| Relation L_1 |
| |
| L_1 = {<x, y, z> in B^3 : x + y + z = 1 mod 2} |
| |
| L_0 contains just the following four |
| triples of the form <x, y, z> in B^3: |
| |
| <0, 0, 1> |
| <0, 1, 0> |
| <1, 0, 0> |
| <1, 1, 1> |
| |
o-------------------------------------------------o
Figure 5. Relation L_1 and Its Plane Projections
In short, the points of L_1, on several projection,
cover all of the points of all three copies of B^2.
In sum, L_0 and L_1 form a 2-projectively indiscernible couplet
of 3-adic relations, establishing the status of each of them as
a 2-projectively irreducible 3-adic relation. QED, I think.
Jon Awbrey
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