Re: Relations And Their Divisitudes

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

RATD.  Note 28

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

Relational Composition as Logical Matrix Multiplication (cont.)

With an eye to extracting a general formula, let us now examine
what we did in multiplying the 2-adic relations G and H together
to obtain their relational composite G o H.

Given the space X = {1, 2, 3, 4, 5, 6, 7}, whose cardinality |X| is 7,
we naturally observe that there are |X x X| = |X| x |X| = 7 x 7 = 49
elementary relations of the form i:j, where i and j range over the
space X.  Although they might be organized in many different ways,
it is convenient to regard the collection of elementary relations
as being arranged in a lexicographic block of the following form:

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

We may think of G and H as being logical sums of the following forms:

   G  =  Sum_ij G_ij (i:j)

   H  =  Sum_ij H_ij (i:j)

The notation "Sum_ij" indicates a logical sum over the collection of
elementary relations i:j, while the factors G_ij, H_ij are values in
the boolean domain B = {0, 1} that are known as the "coefficients"
of the relations G, H, respectively, with regard to each of the
elementary relations i:j in turn.

In general, for a 2-adic relation L, the coefficient L_ij of
the elementary relation i:j in the relation L will be 0 or 1,
respectively, as i:j is excluded from or included in L.

Given all this, we may write out the expansions of G and H as follows:

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

   0(1:1) + 0(1:2) + 0(1:3) + 0(1:4) + 0(1:5) + 0(1:6) + 0(1:7) +
   0(2:1) + 0(2:2) + 0(2:3) + 0(2:4) + 0(2:5) + 0(2:6) + 0(2:7) +
   0(3:1) + 0(3:2) + 0(3:3) + 0(3:4) + 0(3:5) + 0(3:6) + 0(3:7) +
   0(4:1) + 0(4:2) + 1(4:3) + 1(4:4) + 1(4:5) + 0(4:6) + 0(4:7) +
   0(5:1) + 0(5:2) + 0(5:3) + 0(5:4) + 0(5:5) + 0(5:6) + 0(5:7) +
   0(6:1) + 0(6:2) + 0(6:3) + 0(6:4) + 0(6:5) + 0(6:6) + 0(6:7) +
   0(7:1) + 0(7:2) + 0(7:3) + 0(7:4) + 0(7:5) + 0(7:6) + 0(7:7)

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

   0(1:1) + 0(1:2) + 0(1:3) + 0(1:4) + 0(1:5) + 0(1:6) + 0(1:7) +
   0(2:1) + 0(2:2) + 0(2:3) + 0(2:4) + 0(2:5) + 0(2:6) + 0(2:7) +
   0(3:1) + 0(3:2) + 0(3:3) + 1(3:4) + 0(3:5) + 0(3:6) + 0(3:7) +
   0(4:1) + 0(4:2) + 0(4:3) + 1(4:4) + 0(4:5) + 0(4:6) + 0(4:7) +
   0(5:1) + 0(5:2) + 0(5:3) + 1(5:4) + 0(5:5) + 0(5:6) + 0(5:7) +
   0(6:1) + 0(6:2) + 0(6:3) + 0(6:4) + 0(6:5) + 0(6:6) + 0(6:7) +
   0(7:1) + 0(7:2) + 0(7:3) + 0(7:4) + 0(7:5) + 0(7:6) + 0(7:7)

Presenting just the coefficients of G and H on the above plan:

   G  =

   0 0 0 0 0 0 0
   0 0 0 0 0 0 0
   0 0 0 0 0 0 0
   0 0 1 1 1 0 0
   0 0 0 0 0 0 0
   0 0 0 0 0 0 0
   0 0 0 0 0 0 0

   H  =

   0 0 0 0 0 0 0
   0 0 0 0 0 0 0
   0 0 0 1 0 0 0
   0 0 0 1 0 0 0
   0 0 0 1 0 0 0
   0 0 0 0 0 0 0
   0 0 0 0 0 0 0

These are the logical matrix representations of the 2-adic relations G and H.

Jon Awbrey

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