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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some model theory of Polish structures


Author: Krzysztof Krupinski
Journal: Trans. Amer. Math. Soc. 362 (2010), 3499-3533
MSC (2010): Primary 03C45, 03E15; Secondary 54H11, 20E18, 54F15
Published electronically: February 15, 2010
MathSciNet review: 2601598
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Abstract: We introduce a notion of Polish structure and, in doing so, provide a setting which allows the application of ideas and techniques from model theory, descriptive set theory, topology and the theory of profinite groups. We define a topological notion of independence in Polish structures and prove that it has some nice properties. Using this notion, we prove counterparts of some basic results from geometric stability theory in the context of small Polish structures. Then, we prove some structural theorems about compact groups regarded as Polish structures: each small, $ nm$-stable compact $ G$-group is solvable-by-finite; each small compact $ G$-group of finite $ {\mathcal NM}$-rank is nilpotent-by-finite. Examples of small Polish structures and groups are also given.


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Additional Information

Krzysztof Krupinski
Affiliation: Instytut Matematyczny Uniwersytetu Wrocławskiego, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland – and – Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Email: kkrup@math.uni.wroc.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04988-3
PII: S 0002-9947(10)04988-3
Keywords: Polish structure, Polish group, profinite group, independence relation
Received by editor(s): February 11, 2008
Published electronically: February 15, 2010
Additional Notes: This research was supported by the Polish Government grant N201 032 32/2231 and by NSF grant DMS 0300639
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.