CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES
| Autor: | Pantelis E. Eleftheriou |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Group (mathematics)
General Mathematics 010102 general mathematics Structure (category theory) Dimension function Mathematics - Logic Characterization (mathematics) 01 natural sciences Combinatorics Elliptic curve Dimension (vector space) o-minimal structure tame expansion dimension function definable group independent set group chunk Independent set 0103 physical sciences FOS: Mathematics 010307 mathematical physics ddc:510 0101 mathematics Logic (math.LO) Dimension theory (algebra) 03C64 Mathematics |
| Zdroj: | Journal of the Institute of Mathematics of Jussieu. 20:699-724 |
| ISSN: | 1475-3030 1474-7480 |
| DOI: | 10.1017/s1474748019000392 |
| Popis: | We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ 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 ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal. |
| Databáze: | OpenAIRE |
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