SUO: Re: Relations And Their Divisitudes
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RATD. Note 22
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Compositional Analysis of Relations (cont.)
Let's take a moment to be highlight the connections between
two topics that may at first appear to be unrelated, namely:
1. The use of logical conjunction, as denoted by the symbol "&" in
logical expressions of the form "F(x, y, z) = G(x, y) & H(y, z)",
to define a 3-adic relation F by in terms of logical conjunction
and a pair 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 X x Y x Z
via a conjunction of G c X x Y and H c Y x Z, as we may opt
to do by means of an expression that takes on this shape:
F(x, y, z) = G(x, y) & H(y, z).
Nota Bene. Here the equality sign "=" signifies boolean equality,
being thus equivalent to a logical equivalence signified by "<=>".
Visualize the 3-adic relation F c X x Y x Z as a body in XYZ-space,
with G being a figure in XY-space and H being a figure in YZ-space:
o-------------------------------------------------o
| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ /*\ / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
o-------------------------------------------------o
Figure 6. Projections of F onto G and H
If it is true that F(x, y, z) = G(x, y) & H(y, z) for every
point <x, y, z> in the relevant universe of discourse, then
the boolean value of F at the point <x, y, z> is obtainable
by looking at the values of G on <x, y> and H on <y, z> and
then asking whether the point <x, y, z> passes both filters.
That is tantamount to computing F via the following formula:
F(x, y, z) = G(p_12(x, y, z)) & H(p_23(x, y, z)).
In sum, the points of X x Y x Z that satisfy F are just the points
whose projections to X x Y and Y x Z satisfy G and H, respectively.
Logically, that is a conjunction; set-theoretically, intersection.
An intersection of what is what I shall address next on this route.
Jon Awbrey
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