SUO: Re: Examples! Examples! Examples!
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EEE. Note 6
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Example 1. John Sowa's "Top Level Categories" (cont.)
Let's do a survey of the different sorts of lattices and
quotient operations that we can find in the TLC example.
I will leave the language just a bit loose at this point --
we can settle later on what we want to call "the" most
pertinent lattice or quotient in each case. One thing
is clear, though, all of these objects and operations
exist, either in the platonic or sublunar environment,
no matter what we decide to call them at the end of
the day, and so it will be necessary to devote some
attention to each of them, according to the parts
they severally play in understanding the example.
Further, I will continue with the common sort
of casually constructive "models on the cuff"
or "model theory on the installment plan"
that is customary among mathematical and
computer science folks, beginning with
a working knowledge of sets, functions,
relations, categories, and so on, and
giving what passes for constructions
of each and every object or operator
that comes into play.
For the sake of supporting an intuition that
is soon to worked to the max, let's draw some
pictures for the case of an alphabet with just
one term !X!(1) = {x_1}, for convenience using
the nickname x = x_1.
The space X = <|!X!|> = <|x_1|> = <|x|> has 2^1 = 2
cells or positions, pictured in the venn diagram of
Figure 2.
o-------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | | |
| (x) | x | |
| | | |
| o o |
| \ / |
| \ / |
| \ / |
| \ / |
| o-------------o |
| |
| |
o-------------------------------------------------o
Figure 2. One-Dimensional Universe Of Discourse
We can enumerate the 2 positions of X either in
the coordinated form, as X = {<0>, <1>}, or in
the conjunctive form, as X = {(x), x }.
The most obvious lattice that arises from the space X
is the "power set" lattice Pow(X) = 2^X = {W : W c X},
ordered by the set-theoretic inclusion relation "c".
Since this is isomorphic to the proposition lattice
(X^, =>), ordered by the logical implication "=>",
the basic structure of both lattices is captured
by the 2-ptych of 4-gons depicted in Figure 3.
o-------------------------------------o-------------------------------------o
| | |
| X | 1 : X -> B |
| ((u), u } | (( )) |
| @ | @ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| / \ | / \ |
| {(u)} @ @ { u } | (u) @ @ u |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| \ / | \ / |
| @ | @ |
| | ( ) |
| { } | 0 : X -> B |
| | |
o-------------------------------------o-------------------------------------o
Figure 3. Lattice of Subsets Pow(X) and Lattice of Propositions X^ = (X -> B)
Details, Details, ...
Jon Awbrey
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