Publikation: Characterizing o-minimal groups in tame expansions of o-minimal structures
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2021
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Journal of the Institute of Mathematics of Jussieu. Cambridge University Press. 2021, 20(2), pp. 699-724. ISSN 1474-7480. eISSN 1475-3030. Available under: doi: 10.1017/S1474748019000392
Zusammenfassung
We establish the first global results for groups definable in tame expansions of o-minimal structures. Let N be an expansion of an o-minimal structure M that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of M by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if N expands M by a dense independent set, then every definable group is o-minimal.
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510 Mathematik
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o-minimal structure, tame expansion, dimension function, definable group, independent set, group chunk
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ELEFTHERIOU, Pantelis E., 2021. Characterizing o-minimal groups in tame expansions of o-minimal structures. In: Journal of the Institute of Mathematics of Jussieu. Cambridge University Press. 2021, 20(2), pp. 699-724. ISSN 1474-7480. eISSN 1475-3030. Available under: doi: 10.1017/S1474748019000392BibTex
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year={2021},
doi={10.1017/S1474748019000392},
title={Characterizing o-minimal groups in tame expansions of o-minimal structures},
number={2},
volume={20},
issn={1474-7480},
journal={Journal of the Institute of Mathematics of Jussieu},
pages={699--724},
author={Eleftheriou, Pantelis E.}
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<dcterms:abstract xml:lang="eng">We establish the first global results for groups definable in tame expansions of o-minimal structures. Let N be an expansion of an o-minimal structure M that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of M by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if N expands M by a dense independent set, then every definable group is o-minimal.</dcterms:abstract>
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