Re: Relations And Their Divisitudes

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RATD.  Note 24

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Compositional Analysis of Relations (cont.)

Let's now look at a way of defining the relational composition
of 2-adic relations by exploiting the set-theoretic operational
resources of intersections, projections, and tacit extensions.
To be specific, I will define the relational composition of a
couple of 2-adic relations in terms of their respective tacit
extensions to 3-adic relations, followed by the set-theoretic
intersection of these two tacit extensions, and then by the
projection of this intersection, which happens to be the
largest 3-adic relation that fits the prima facie 2-adic
projection data, into a third 2-adic relation, which is
the thus-computed relational composition of the given
pair of 2-adic relations.

I usually think of this definition of composition as "Tarski's Trick",
because I learned it from a paper of Ulam that attributes it to Tarski,
but I would not be terribly surprised if I suddenly recognized the same
idea in the work of Peirce, DeMorgan, or even Newton, for that matter.

| Ulam, S.M. & Bednarek, A.R.,
|"On the Theory of Relational Structures and Schemata for Parallel Computation",
| in Ulam & Bednarek (eds.), 'ABA', pp. 477-508, report dated 1977.
|
| Ulam, F. & Bednarek, A.R. (eds.),
|'Analogies Between Analogies:
| The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators',
| University of California Press, Berkeley, 1990.

We begin with a pair of 2-adic relations G, H c X x Y.

o-------------------------------------------------o
|                                                 |
|        o                       o                |
|        |\                      |\               |
|        | \                     | \              |
|        |  \                    |  \             |
|        |   \                   |   \            |
|        |    \                  |    \           |
|        |     \                 |     \          |
|        |   *  \                |   *  \         |
|        X   *   Y               X   *   Y        |
|         \  *   |                \  *   |        |
|          \ G   |                 \ H   |        |
|           \    |                  \    |        |
|            \   |                   \   |        |
|             \  |                    \  |        |
|              \ |                     \ |        |
|               \|                      \|        |
|                o                       o        |
|                                                 |
o-------------------------------------------------o
Figure 10.  Dyadic Relations G, H c X x Y

Mark that H is not exactly the same H that we had before,
because this H is presented in the same plane X x Y as G.
Whether you view isomorphic things to be the same things
or not, you still have to specify the exact isomorphisms
that are needed to transform any given representation of
a thing into a required representation of the same thing.
Let us imagine that we have done this, and say how later:

o-------------------------------------------------o
|                                                 |
|        o                               o        |
|        |\                             /|        |
|        | \                           / |        |
|        |  \                         /  |        |
|        |   \                       /   |        |
|        |    \                     /    |        |
|        |     \                   /     |        |
|        |   *  \                 /  *   |        |
|        X   *   Y               Y   *   Z        |
|         \  *   |               |   *  /         |
|          \ G   |               |   H'/          |
|           \    |               |    /           |
|            \   |               |   /            |
|             \  |               |  /             |
|              \ |               | /              |
|               \|               |/               |
|                o               o                |
|                                                 |
o-------------------------------------------------o
Figure 11.  Dyadic Relations G c X x Y and H' c Y x Z

That will do for a setup.

Jon Awbrey

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