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, it's a conjunction;  set-theoretically, an intersection.
An intersection of what is what I shall address next on this route.

Jon Awbrey

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