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Re: This Week's Finds in Mathematical Physics (Week 259)
Posted:
Feb 29, 2008 8:15 AM
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Hello,
John Baez wrote:
> In article <475d57b1$0$502$b45e6eb0@senator-bedfellow.mit.edu>,
> <tchow@lsa.umich.edu> wrote:
>
>>Another way to understand the analogy between sets and vector spaces
>>is via the language of matroids. Sets correspond to "free matroids"
>>and vector spaces correspond to "representable matroids." How much
>>of what you say about the one-element field can be understood in
>>terms of matroids?
>
> I don't know, since I don't know much about matroids. But, from
> what little I know, a lot of projective geometry fits nicely into
> the matroid framework. That's promising, since the stuff I'm
> talking about is closely related to projective geometry.
>
> According to what you say, both for free matroids and "representable
> matroids" (whatever those are) over a given field we get one of these,
> up to isomorphism, for each natural number n - the cardinality or
> dimension. Are there other nice series of matroids, one for each
> natural number? That might lead to some fun stuff.
The following reference might contain an explanation of your
observation:
B. Zilber, Uncountably categorical theories. Transl. from the Russian
by D. Louvish. Transl. ed. by Simeon Ivanov. (English) Translations of
Mathematical Monographs. 117. Providence, RI: American Mathematical
Society. vi, 122 p. (1993). ISBN 0-8218-4586-1/hbk
One of the main results is the following:
Trichotomy Theorem: Every uncountably categorical theory is of one of
the following types, according to the geometries it realises:
(1) field-like
(2) module-like
(3) disintegrated
Examples are:
(1) algebraically closed fields
(2) vector spaces
(3) sets
Achim
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Achim Blumensath \O/ \___/\ |
TU Darmstadt =o= \ /\ \|
www.mathematik.tu-darmstadt.de/~blumensath /"\ o----|
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