Groupes Fins

Type of Publication: Journal article
URI (citable link): http://nbn-resolving.de/urn:nbn:de:bsz:352-0-280478
Author: Milliet, Cedric
Year of publication: 2014
Published in: The Journal of Symbolic Logic ; 79 (2014), 4. - pp. 1120-1132. - ISSN 0022-4812. - eISSN 1943-5886
DOI (citable link): https://dx.doi.org/10.1017/jsl.2014.12
Summary:
We investigate some common points between stable structures and weakly small structures and define a structure M to be fine if the Cantor-Bendixson rank of the topological space Sφ(dcleq(A)) is an ordinal for every finite subset A of M and every formula φ(x.y) where x is of arity 1. By definition, a theory is fine if all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subset A of a fine group G, the traces on the algebraic closure acl(A) of A of definable subgroups of G over acl(A) which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions.
Subject (DDC): 510 Mathematics
Keywords: Model theory, Cantor-Bendixson rank, local descending chain condition, Artin-Schreier extension
Link to License: Terms of use
Bibliography of Konstanz: Yes
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MILLIET, Cedric, 2014. Groupes Fins. In: The Journal of Symbolic Logic. 79(4), pp. 1120-1132. ISSN 0022-4812. eISSN 1943-5886. Available under: doi: 10.1017/jsl.2014.12

@article{Milliet2014Group-30526, title={Groupes Fins}, year={2014}, doi={10.1017/jsl.2014.12}, number={4}, volume={79}, issn={0022-4812}, journal={The Journal of Symbolic Logic}, pages={1120--1132}, author={Milliet, Cedric} }

2015-03-24T15:22:52Z Groupes Fins 2014 eng Milliet, Cedric Milliet, Cedric terms-of-use We investigate some common points between stable structures and weakly small structures and define a structure M to be fine if the Cantor-Bendixson rank of the topological space Sφ(dcl<sup>eq</sup>(A)) is an ordinal for every finite subset A of M and every formula φ(x.y) where x is of arity 1. By definition, a theory is fine if all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subset A of a fine group G, the traces on the algebraic closure acl(A) of A of definable subgroups of G over acl(A) which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions. 2015-03-24T15:22:52Z

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