Re: Relations And Their Divisitudes

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

RATD.  Note 31

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Relational Composition as Logical Matrix Multiplication (cont.)

There is another form of representation for 2-adic relations that
is useful to keep in mind, especially for its ability to render the
logic of many complex formulas almost instantly understandable to the
mind's eye.  This is the representation in terms of "bipartite graphs",
or "bigraphs" for short.

Here is what G and H look like in the bigraph picture:

o---------------------------------------o
|                                       |
|         1   2   3   4   5   6   7     |
|         o   o   o   o   o   o   o     |
|                    /|\                |
|    G              / | \               |
|                  /  |  \              |
|         o   o   o   o   o   o   o     |
|         1   2   3   4   5   6   7     |
|                                       |
o---------------------------------------o
Figure 14.  G = 4:3 + 4:4 + 4:5

o---------------------------------------o
|                                       |
|         1   2   3   4   5   6   7     |
|         o   o   o   o   o   o   o     |
|                  \  |  /              |
|    H              \ | /               |
|                    \|/                |
|         o   o   o   o   o   o   o     |
|         1   2   3   4   5   6   7     |
|                                       |
o---------------------------------------o
Figure 15.  H = 3:4 + 4:4 + 5:4

These graphs may be read to say:
G puts 4 in relation to 3, 4, 5.
H puts 3, 4, 5 in relation to 4.

To form the composite relation G o H, we simply follow the bigraph for G
by the bigraph for H, here arranging the bigraphs in order down the page,
and then we proceed to "squeeze out the middle man", that is, we call any
non-empty set of paths of length two between two nodes as being equivalent
to a single directed edge between them in the composite bigraph for G o H.

Here's how it looks in pictures:

o---------------------------------------o
|                                       |
|         1   2   3   4   5   6   7     |
|         o   o   o   o   o   o   o     |
|                    /|\                |
|    G              / | \               |
|                  /  |  \              |
|         o   o   o   o   o   o   o     |
|                  \  |  /              |
|    H              \ | /               |
|                    \|/                |
|         o   o   o   o   o   o   o     |
|         1   2   3   4   5   6   7     |
|                                       |
o---------------------------------------o
Figure 16.  G Followed By H

o---------------------------------------o
|                                       |
|         1   2   3   4   5   6   7     |
|         o   o   o   o   o   o   o     |
|                     |                 |
|  G o H              |                 |
|                     |                 |
|         o   o   o   o   o   o   o     |
|         1   2   3   4   5   6   7     |
|                                       |
o---------------------------------------o
Figure 17.  G Composed With H

Once again we find that G o H = 4:4.

Most likely you are remembering
of what I said about redundancy.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o