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From the cosmic urn of all possible functions
we draw the unique random function f : X -> Y.
o-----------------------------o-----------------------------o
| Source Domain | Target Codomain |
o-----------------------------o-----------------------------o
| |
| x_1 o--------------------------· o y_1 |
| \ |
| \ |
| x_2 o-----------------------------o y_2 |
| / |
| / |
| x_3 o--------------------------· o y_3 |
| |
| |
| x_4 o--------------------------· o y_4 |
| \ |
| \ |
| o y_5 |
| / |
| / |
| x_5 o--------------------------· o y_6 |
| |
| |
| Functional Fibers |
o-----------------------------------------------------------o
Now we form the canonical decomposition or factorization of f
into a surjective function followed by an injective function.
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 |
| |
| |
| Functional Factors |
o-----------------------------------------------------------o
Here we have the following data:
f = h o g
f : X -> Y, arbitrary
g : X -> M, surjective
h : M -> Y, injective
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}
f = {<x_1, y_2>, <x_2, y_2>, <x_3, y_2>, <x_4, y_5>, <x_5, y_5>}
g = {<x_1, m_1>, <x_2, m_1>, <x_3, m_1>, <x_4, m_2>, <x_5, m_2>}
h = {<m_1, y_2>, <m_2, y_5>}
I think that you can see
from this sufficient bit
that it lies in the very
nature of a function for
it factorizing to permit.
Top O The Morning,
From Middle Earth!
Jon Awbrey
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