ONT Re: Reductions Among Relations
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RAR. Note 15
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Compositional Analysis of Relations (cont.)
Let us now look at a way of defining the relational composition of
2-adic relations by using the set-theoretic operational resources
of intersections, projections, and tacit extensions. To be more
specific, we will define the relational composition of a couple
of 2-adic relations in terms of their separate tacit extensions
to 3-adic relations, followed by the set intersection of these
tacit extensions, and then the projection of this intersection,
tantamount to the maximal 3-adic relation that is consistent
with the 'prima facie' 2-adic relational data, into a third
2-adic relation, the computed composition of the first two.
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 suddenly to recognize it in 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.
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| |
| o o |
| |\ |\ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | \ | \ |
| | * \ | * \ |
| X * Y X * Y |
| \ * | \ * | |
| \ G | \ H | |
| \ | \ | |
| \ | \ | |
| \ | \ | |
| \ | \ | |
| \| \| |
| o o |
| |
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Figure 5. 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:
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| |
| o o |
| |\ /| |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | * \ / * | |
| X * Y Y * Z |
| \ * | | * / |
| \ G | | H'/ |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
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Figure 6. Dyadic Relations G c X x Y and H' c Y x Z
Jon Awbrey
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