SUO: Re: Examples! Examples! Examples!
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EEE. Note 7
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Example 1. John Sowa's "Top Level Categories" (cont.)
Generally speaking, if expressing it very roughly, a quotient operation
is the act of ignoring certain distinctions that might otherwise be made.
For example, given the integers Z = {..., -2, -1, 0, 1, 2, ...}, taking
the "integers modulo 2", written Z_2 ~=~ {0, 1}, amounts to neglecting
the distinctions that are otherwise made between the members of the set
2Z = {..., -4, -2, 0, 2, 4, ...}, commonly known as "even integers",
and consequently collapsing the distinctions that are normally made
between the members of the set 2Z + 1 = {..., -3, -1, 1, 3, ...},
commonly known as "odd integers", and thereby resulting in but
a single distinction, that between the "equivalence classes"
or the "residue classes" of Evens and Odds, respectively.
Drawing on the primer of the 1-dimensional universe X% = [x], we may
derive some clue what the spaces Pow(A) and A^ look like as lattices:
Pow(A) has the set A at the top and the empty set {} at the bottom,
while A^ has the constantly true proposition (()) = 1 : A -> B at
the top and the constantly false proposition () = 0 : A -> B at
the bottom. Two down, 2^(2^25) - 2 to go! Which goes a long way
to explain why we find the delicately balanced opposition between
generators and relations to be an essential tension of the subject.
Jon Awbrey
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