ONT Re: Logic Of Relatives
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
LOR. Note 56
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
First, a correction. Ignore for now the
gloss that I gave in regard to Figure 19:
| Here, I have used arrowheads to indicate the relational domains
| at which each of the relations J, K, L happens to be functional.
It is more like the feathers of the arrows that serve to mark the
relational domains at which the relations J, K, L are functional,
but it would take yet another construction to make this precise,
as the feathers are not uniquely appointed but many splintered.
Now, as promised, let's look at a more homely example of a morphism,
say, any one of the mappings J : R -> R (roughly speaking) that are
commonly known as "logarithm functions", where you get to pick your
favorite base. In this case, K(r, s) = r + s and L(u, v) = u . v,
and the defining formula J(L(u, v)) = K(Ju, Jv) comes out looking
like J(u . v) = J(u) + J(v), writing a dot (.) and a plus sign (+)
for the ordinary 2-ary operations of arithmetical multiplication
and arithmetical summation, respectively.
o-----------------------------------------------------------o
| |
| {+} {.} |
| @ @ |
| /|\ /|\ |
| / | \ / | \ |
| v | \ v | \ |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| ^ ^ ^ / / / |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| @ @ @ |
| J J J |
| |
o-----------------------------------------------------------o
Figure 21. Logarithm Arrow J : {+} <- {.}
Thus, where the "image" J is the logarithm map,
the "compound" K is the numerical sum, and the
the "ligature" L is the numerical product, one
obtains the immemorial mnemonic motto:
| The image of the product is the sum of the images.
|
| J(u . v) = J(u) + J(v)
|
| J(L(u, v)) = K(Ju, Jv)
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o