SUO: Re: Relations And Their Divisitudes
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RATD. Note 32
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Relational Composition as Logical Matrix Multiplication (concl.)
We have now seen three different representations of 2-adic relations.
If one has a strong preference for letters, or numbers, or pictures,
then one may be tempted to take one or the other as being canonical,
but each of them will be found to have its peculiar advantages and
disadvantages in any given application, and the maximum advantage
is therefore approached by keeping all three of them in mind.
To see the promised utility of the bigraph picture for 2-adic relations,
let us devise a slightly more complex example of a composition problem,
using it to illustrate the logic of the matrix multiplication formula.
Keeping to the same space X = {1, 2, 3, 4, 5, 6, 7},
define the 2-adic relations M, N c X x X as follows:
M = 2:1 + 2:2 + 2:3 + 4:3 + 4:4 + 4:5 + 6:5 + 6:6 + 6:7
N = 1:1 + 2:1 + 3:3 + 4:3 + 4:5 + 5:5 + 6:7 + 7:7
Here are the bigraph pictures:
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o |
| /|\ /|\ /|\ |
| M / | \ / | \ / | \ |
| / | \ / | \ / | \ |
| o o o o o o o |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 18. Dyadic Relation M
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o |
| | / | / \ | \ | |
| N | / | / \ | \ | |
| |/ |/ \| \| |
| o o o o o o o |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 19. Dyadic Relation N
To form the composite relation M o N, we simply follow the bigraph for M
by the bigraph for N, here arranging the bigraphs in order down the page,
and then we proceed to "edge out the middle person", 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 M o N.
Here's how it looks in pictures:
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o |
| /|\ /|\ /|\ |
| M / | \ / | \ / | \ |
| / | \ / | \ / | \ |
| o o o o o o o |
| | / | / \ | \ | |
| N | / | / \ | \ | |
| |/ |/ \| \| |
| o o o o o o o |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 20. M Followed By N
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o |
| / \ / \ / \ |
| M o N / \ / \ / \ |
| / \ / \ / \ |
| o o o o o o o |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 21. M Composed With N
Let us hark back to that mysterious matrix multiplication formula,
and see how it appears in the light of the bigraph representation.
The coefficient of the composition M o N
between i and j in X is given as follows:
(M o N)_ij = Sum_k (M_ik N_kj)
Graphically interpreted, this is a "sum over paths".
Starting at the node i, M_ik being 1 indicates that
there is an edge in the bigraph of M from node i to
node k, and N_kj being 1 indicates that there is an
edge in the bigraph of N from node k to node j. So
the Sum_k ranges over all possible intermediaries k,
ascending from 0 to 1 just as soon as there happens
to be some path of length two between nodes i and j.
Incidentally -- or non-incidentally, as you will see --
it is instructive here to compute the other possible
composition that can be made of these two components,
namely, the composition N o M, that takes M and N in
the opposite order. Here is the graphic computation:
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o |
| | / | / \ | \ | |
| N | / | / \ | \ | |
| |/ |/ \| \| |
| o o o o o o o |
| /|\ /|\ /|\ |
| M / | \ / | \ / | \ |
| / | \ / | \ / | \ |
| o o o o o o o |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 22. N Followed By M
o---------------------------------------o
| |
| 1 2 3 4 5 6 7 |
| o o o o o o o |
| |
| N o M |
| |
| o o o o o o o |
| 1 2 3 4 5 6 7 |
| |
o---------------------------------------o
Figure 23. N Composed With M
What this says about NoMinal thinking
I leave as an exorcise for the reader,
but it gives sufficient evidence that
relational composition, just like its
kindred, matrix multiplication, forms
a non-commutative algebraic operation.
Jon Awbrey
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