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Directory:Jon Awbrey/Notes/Factorization Issues (view source)
Revision as of 22:00, 24 June 2009
, 22:00, 24 June 2009→Note 1: markup
Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''. Here is one picture of an <math>f : X \to Y,</math> just about as generic as it needs to be:
Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''. Here is one picture of an <math>f : X \to Y,</math> just about as generic as it needs to be:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
</pre>
</pre>
|}
Now, it is a fact that any old function that you might pick ''factors'' into a surjective ("onto") function and an injective ("one-to-one") function, in the present example just like so:
Now, it is a fact that any old function that you might pick ''factors'' into a surjective ("onto") function and an injective ("one-to-one") function, in the present example just like so:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
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<pre>
<pre>
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
</pre>
</pre>
|}
Writing the functional compositions <math>f = g \circ h</math> "on the right", as they say, we have the following data about the situation:
Writing the functional compositions <math>f = g \circ h</math> "on the right", as they say, we have the following data about the situation:
signs, like "=", written between ostensible nodes,
signs, like "=", written between ostensible nodes,
like "o", identify them into a single real node.
like "o", identify them into a single real node.
</pre>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
o-----------------------------o
o-----------------------------o
| Denotative Component of L |
| Denotative Component of L |
| |
| |
o-----------------------------o
o-----------------------------o
</pre>
|}
<pre>
This depicts a situation where each of the three objects,
This depicts a situation where each of the three objects,
x_1, x_2, x_3, has a "proper name" that denotes it alone,
x_1, x_2, x_3, has a "proper name" that denotes it alone,
In this case, we factor the function f : O -> S
In this case, we factor the function f : O -> S
</pre>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
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<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| |
| |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
into the composition g o h, where g : O -> M, and h : M -> S
into the composition g o h, where g : O -> M, and h : M -> S
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| |
| |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
<pre>
The factorization of an arbitrary function
The factorization of an arbitrary function
into a surjective ("onto") function followed
into a surjective ("onto") function followed
we get the augmented sign relation L', shown
we get the augmented sign relation L', shown
in the next vignette:
in the next vignette:
</pre>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<pre>
o-----------------------------o
o-----------------------------o
| Denotative Component of L' |
| Denotative Component of L' |
| |
| |
o-----------------------------o
o-----------------------------o
</pre>
|}
<pre>
This amounts to the creation of a hypostatic object x,
This amounts to the creation of a hypostatic object x,
which affords us a singular denotation for the sign y.
which affords us a singular denotation for the sign y.