ONT Re: Zeroth Order Ontology
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ZOO. Note 9
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Generators and Relations.
I have just written roughshod over a number of fine points
that would amount to sources of concern for graph theorists
but constitute matters of indifference for the sake of logic,
and so I will silently pass them by, resting content with the
knowledge that I have already given better accounts elsewhere.
This roughshod character could always be refined by interposing
another layer of equivalence relarions, and so I will leave that
as an exercise for the reader's imagination.
By way of introducing a few more bits of useful terminology, let's
now compare and constrast the two quotient structures that we have
been considering up to this point.
Let L_1 = "tally marks mod 2".
Let L_2 = "rooted trees mod I_1, I_2".
Given a couple of typical elements of L_1, say "//////" and "///////",
we apply the relation "//" = "" three times to each to arrive at the
equivalences "//////" = "" and "///////" = "/".
We say that the space L_1 partitions under the
relation "//" = "" into the following "cosets":
Even = {"", "//", "////", "//////", ...}
Odd = {"/", "///", "/////", "///////", ...}
The empty string !e! = "" and the single mark "/" are evidently the
simplest members of their respective cosets, and so it is customary
to refer to them as "canonical forms" or "canonical representatives"
of these cosets, respectively. One uses square brackets around any
element of a coset to denote the coset of that element, for example:
Even = [""] = ["//"] = ["////"] = ...
Odd = ["/"] = ["///"] = ["/////"] = ...
To be continued ...
Jon Awbrey
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