SUO: Re: Examples! Examples! Examples!
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EEE. Note 15
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Here is a way to visualize two sorts of lattice quotients, operating in
the syntactic space !C!(x) of sentences involving a single variable "x"
and the objective space X^ = (X -> B) of propositions that are intended
as the denotations of these expressions, respectively. For simplicity,
I have shown how these operations look over the 1-dimensional universe.
Table 9-1a shows a sample of expressions in the cactus language !C!(x),
namely, with labels from the palette of one paint !P! = {p_1} = {"x"}.
The formal language !C!(x) is just an amorphous set of sentences, but
if we partition it into logical equivalence classes by applying the
relevant set of axioms $A$(=) for logical equivalence (<=>), then
the result !C!(x)/$A$(=) can be organized as a lattice structure.
Table 9-1b shows the result of asserting or assuming the proposition x,
that is, imposing the relation "x = 1" or "x is true". This amounts to
coalescing the logical equivalence classes even further, tantamount to
pretending that x is an alias for the blank space that signifies truth.
Table 9-2 shows the images of these actions in the corresponding
objective spaces, tracing the quotient mapping from X^ to X^/x
that identifies x with 1 and (x) with 0.
Table 9-1. Syntactic Quotient Lattices
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| a. !C!(x)/$A$(=) | b. C!{x)/($A$(=), x} |
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| | |
| o-----------o | o-----------o |
| /| " " |= | | " " | |
| / | "(x(x))" | = | | "(x(x))" | |
| / | ... | = | | ... | |
| / o-----------o = | | "x" | |
| / = | | "((x))" | |
| / = | | ... | |
| o-----------o o-----------o | o-----o-----o |
| | "(x)" | | "x" | | | |
| | "(x(()))" | | "((x))" | | | |
| | ... | | ... | | | |
| o-----------o o-----------o | o-----o-----o |
| = / | | "()" | |
| = / | | "x(x)" | |
| = o-----------o / | | ... | |
| = | "()" | / | | "(x)" | |
| = | "x(x)" | / | | "(x(()))" | |
| =| ... |/ | | ... | |
| o-----------o | o-----------o |
| | |
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Table 9-2. Objective Quotient Lattices
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| a. X^ ~=~ 2^X | b. X^/x |
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| | |
| | x |
| | |
| o | o |
| / = | | |
| / = | | |
| / = | | |
| / = | | |
| / = | | |
| / = | | |
| (x) o o x | | |
| = / | | |
| = / | | |
| = / | | |
| = / | | |
| = / | | |
| = / | | |
| o | o |
| | |
| ( ) | (x) |
| | |
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Jon Awbrey
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