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Summary: UIC Model Theory Seminar, February 24, 2004
Note special time: 2pm in 427 SEO
Theories with a finite number of countable models
and group polygonometries
Sergey Sudoplatov, Novosibirsk State Technical University
We introduce the classification of complete countable first-order theories
with a finite number of pairwise non-isomorphic countable models relative
to the two main features: the Rudin-Keisler preorder and the distribution
function of the number of limiting models over types. Herewith the main
(nontrivial) part of the classification spreads on to Ehrenfeuchtian theories,
i.e. non omega-categorical theories with a finite number of countable models.
We generalize the classification to an arbitrary case of finite Rudin-Keisler
preorder. We show that the same features play the key role in this case, and
prove the consistency of any finite Rudin-Keisler preorder with an arbitrary
distribution function f, satisfying the condition rangf {, 2}. The
notion of powerful digraph is defined and it is shown that the presence and
the structure of a powerful digraph in the structure of nonisolated powerful
type plays the defining role in the construction of Ehrenfeuchtian theories.
The notions of group polygonometry and group trigonometry are defined
and it is shown that the structure of any acyclic group trigonometry on a
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