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Topic: This Week's Finds in Mathematical Physics (Week 259)
Replies: 4   Last Post: Feb 29, 2008 8:15 AM

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Achim Blumensath

Posts: 10
Registered: 6/21/05
Re: This Week's Finds in Mathematical Physics (Week 259)
Posted: Feb 29, 2008 8:15 AM
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John Baez wrote:
> In article <475d57b1$0$502$>,
> <> 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

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 Blumensath \O/ \___/\ |
TU Darmstadt =o= \ /\ \| /"\ o----|

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