ONT Re: Reductions Among Relations
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RAR. Note 13
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Compositional Analysis of Relations (cont.)
Before we continue with the analysis of the Between relation, let us
take a moment to make sure that we understand the connections between
two topics that may appear at first to be entirely unrelated, namely:
1. A certain use of the logical conjunction, denoted by "&", as it appears
in logical expressions of the form "F(x, y, z) = G(x, y) & H(y, z)",
and that we use to define a 3-adic relation F by means of this "&"
and in terms of a couple of 2-adic relations G and H.
2. The concepts of 2-adic "projection" and "projective determination",
that are invoked in the "weak" notion of "projective reducibility".
Let us begin by drawing ourselves a picture of what is really going on whenever
we formulate a definition of F c XxYxZ via a conjunction of G c XxY and H c YxZ,
as we may choose to do by means of an expression of the following form:
F(x, y, z) = G(x, y) & H(y, z).
Visualize the 3-adic relation F c XxYxZ as a body in XYZ-space,
while G is a figure in XY-space and H is a figure in YZ-space:
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| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ *** / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
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Figure 1. Projections of F onto G and H
Jon Awbrey
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