ONT Re: Extension x Comprehension = Information
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
I think that we have fairly well convinced ourselves --
at least, I am reasonably sure that some of us have --
that every function can be factored into an "onto"
followed by a "one-to-one" mapping, as shown here:
o-------------------o-------------------o-------------------o
| Source Domain | Middle Domain | Target Domain |
o-------------------o---------------------------------------o
| |
| x_1 o-----------· o y_1 |
| \ |
| \ m_1 |
| x_2 o--------------o------------->o y_2 |
| / |
| / |
| x_3 o-----------· o y_3 |
| |
| |
| x_4 o-----------· o y_4 |
| \ |
| \ m_2 |
| o------------->o y_5 |
| / |
| / |
| x_5 o-----------· o y_6 |
| |
| |
| Factured Fiber Trails |
o-----------------------------------------------------------o
So patent is the pending of Damocles' Razor on our modern incre-mentalities
that we would scarcely dare to think of it this way without a little bit of
prodding, but it is possible to treat this functional fractionation process
as a case of transmuting a 2-adic relation f c X x Y into a 3-adic relation
L c X x M x Y.
In our present example we have the data:
| f c X x Y
|
| X = {x_1, x_2, x_3, x_4, x_5}
|
| Y = {y_1, y_2, y_3, y_4, y_5, y_6}
|
| f = {<x_1, y_2>,
| <x_2, y_2>,
| <x_3, y_2>,
| <x_4, y_5>,
| <x_5, y_5>}
and
| L c X x M x Y
|
| X = {x_1, x_2, x_3, x_4, x_5}
|
| M = {m_1, m_2}
|
| Y = {y_1, y_2, y_3, y_4, y_5, y_6}
|
| L = {<x_1, m_1, y_2>,
| <x_2, m_1, y_2>,
| <x_3, m_1, y_2>,
| <x_4, m_2, y_5>,
| <x_5, m_2, y_5>}
I will let you stare
at that for a while,
and so will I.
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤