[Inquiry] Re: Reductions Among Relations

[Jon Awbrey]

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RAR.  Note 12

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3.  Compositional Analysis of Relations (cont.)

Let us now render the picture of our composition example
a little less impressionistic and a little more realistic
in the manner of its representation, and let us accomplish
this through the introduction of coordinates, in other words,
concrete names for the objects that we relate through various
forms of relations, 2-adic and 3-adic in the present instance.

Revising the Example along these lines
would give a Figure like the following:

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o       o       o                |
|                |\     / \     /|                |
|                | \   / F \   / |                |
|                |  \ /  *  \ /  |                |
|                |   \  /*\  /   |                |
|                |  / \//*\\/ \  |                |
|                | /  /\/ \/\  \ |                |
|                |/  ///\ /\\\  \|                |
|        o       X  ///  Y  \\\  Z       o        |
|        |\      7\///   |   \\\/7      /|        |
|        | \      6//    |    \\6      / |        |
|        |  \    //5\    |    /5\\    /  |        |
|        |   \  /// 4\   |   /4 \\\  /   |        |
|        |    \///   3\  |  /3   \\\/    |        |
|        |   G/\/     2\ | /2     \/\H   |        |
|        |   *//\      1\|/1      /\\*   |        |
|        X   *\  Y       o       Y  /*   Z        |
|        7\  *\\ |7             7| //*  /7        |
|         6\ |\\\|6             6|///| /6         |
|          5\| \\@5             5@// |/5          |
|           4@  \@4             4@/  @4           |
|            3\  @3             3@  /3            |
|             2\ |2             2| /2             |
|              1\|1             1|/1              |
|                o               o                |
|                                                 |
o-------------------------------------------------o
Figure 7.  F as the Intersection of TE(G) and TE(H)

By way of the representation that is accorded us by these coordinates,
we have the following data with regard to F c XxYxZ, G c XxY, H c YxZ.

F  =  4:3:4  +  4:4:4  +  4:5:4

G  =  4:3    +  4:4    +  4:5

H  =    3:4  +    4:4  +    5:4

Let us now verify that all of the proposed definitions, formulas, and other
relationships check out against the concrete data of the composition example.
The ultimate goal is to develop a clearer picture of what is going on in the
formula that expresses the relational composition in terms of the projection
of the intersection of the tacit extensions:

Let us now verify that all of the proposed definitions,
formulas, and other relationships check out against the
concrete data of the composition example.  The ultimate
goal is to develop a clearer picture of what is going on
in the formula that expresses the relational composition
of 2-adic relations in terms of the extremal projection
of the intersection of their tacit 3-adic extensions:

G o H  =  Proj_XZ (TE_XY_Z (G) |^| TE_YZ_X (H)).

Here is the big picture, with all of the pieces:

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   /         \                   |
|                  /           \                  |
|                 /    G o H    \                 |
|                X       *       Z                |
|                7\     /|\     /7                |
|                 6\   / | \   /6                 |
|                  5\ /  |  \ /5                  |
|                   4@   |   @4                   |
|                    3\  |  /3                    |
|                     2\ | /2                     |
|                      1\|/1                      |
|                        |                        |
|                        |                        |
|                        |                        |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o       |       o                |
|                |\     /|\     /|                |
|                | \   / F \   / |                |
|                |  \ /  *  \ /  |                |
|                |   \  /*\  /   |                |
|                |  / \//*\\/ \  |                |
|                | /  /\/ \/\  \ |                |
|                |/  ///\ /\\\  \|                |
|        o       X  ///  Y  \\\  Z       o        |
|        |\       \///   |   \\\/       /|        |
|        | \      ///    |    \\\      / |        |
|        |  \    ///\    |    /\\\    /  |        |
|        |   \  ///  \   |   /  \\\  /   |        |
|        |    \///    \  |  /    \\\/    |        |
|        |   G/\/      \ | /      \/\H   |        |
|        |   *//\       \|/       /\\*   |        |
|        X   *\  Y       o       Y  /*   Z        |
|        7\  *\\ |7             7| //*  /7        |
|         6\ |\\\|6             6|///| /6         |
|          5\| \\@5             5@// |/5          |
|           4@  \@4             4@/  @4           |
|            3\  @3             3@  /3            |
|             2\ |2             2| /2             |
|              1\|1             1|/1              |
|                o               o                |
|                                                 |
o-------------------------------------------------o
Figure 8.  G o H  =  Proj_XZ (TE(G) |^| TE(H))

All that remains to do now is to check the following
collection of data and derivations against Figure 8.

F  =  4:3:4  +  4:4:4  +  4:5:4

G  =  4:3    +  4:4    +  4:5

H  =    3:4  +    4:4  +    5:4

G o H  =  (4:3 + 4:4 + 4:5)(3:4 + 4:4 + 5:4)

       =  4:4

TE(G)  =  TE_XY_Z (G)

       =  Sum_z=1...7 (4:3:z + 4:4:z + 4:5:z)

       =  4:3:1 + 4:4:1 + 4:5:1 +
          4:3:2 + 4:4:2 + 4:5:2 +
          4:3:3 + 4:4:3 + 4:5:3 +
          4:3:4 + 4:4:4 + 4:5:4 +
          4:3:5 + 4:4:5 + 4:5:5 +
          4:3:6 + 4:4:6 + 4:5:6 +
          4:3:7 + 4:4:7 + 4:5:7

TE(H)  =  TE_YZ_X (H)

       =  Sum_x=1...7 (x:3:4 + x:4:4 + x:5:4)

       =  1:3:4 + 1:4:4 + 1:5:4 +
          2:3:4 + 2:4:4 + 2:5:4 +
          3:3:4 + 3:4:4 + 3:5:4 +
          4:3:4 + 4:4:4 + 4:5:4 +
          5:3:4 + 5:4:4 + 5:5:4 +
          6:3:4 + 6:4:4 + 6:5:4 +
          7:3:4 + 7:4:4 + 7:5:4

TE(G) |^| TE(H)  =  4:3:4 + 4:4:4 + 4:5:4

G o H  =  Proj_XZ (TE(G) |^| TE(H))

       =  Proj_XZ (4:3:4 + 4:4:4 + 4:5:4)

       =  4:4

By my lights, anyway, it all checks.

Jon Awbrey

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