Journal of Symbolic Logic 86 (2):531-557 (2021)
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In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note.
For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation.
The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of stp(a) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup.
Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of stp(ac) is abelian, nor the assumption of c being finite can be removed.
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DOI | 10.1017/jsl.2021.31 |
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References found in this work BETA
Galois Groups of First Order Theories.E. Casanovas, D. Lascar, A. Pillay & M. Ziegler - 2001 - Journal of Mathematical Logic 1 (02):305-319.
The Lascar Groups and the First Homology Groups in Model Theory.Jan Dobrowolski, Byunghan Kim & Junguk Lee - 2017 - Annals of Pure and Applied Logic 168 (12):2129-2151.
Boundedness and Absoluteness of Some Dynamical Invariants in Model Theory.Krzysztof Krupiński, Ludomir Newelski & Pierre Simon - 2019 - Journal of Mathematical Logic 19 (2):1950012.
Classifying Torsion Free Groups in o-Minimal Expansions of Real Closed Fields.Eliana Barriga & Alf Onshuus - 2016 - Annals of Pure and Applied Logic 167 (12):1267-1297.
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