The relativized Lascar groups, type-amalgamation, and algebraicity

Journal of Symbolic Logic 86 (2):531-557 (2021)
  Copy   BIBTEX

Abstract

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.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,931

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

The Lascar Group and the Strong Types of Hyperimaginaries.Byunghan Kim - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):497-507.
A note on Lascar strong types in simple theories.Byunghan Kim - 1998 - Journal of Symbolic Logic 63 (3):926-936.
A note on Lascar strong types in simple theories.Byunghan Kim - 1998 - Journal of Symbolic Logic 63 (3):926-936.
Simple groups and the number of countable models.Predrag Tanović - 2013 - Archive for Mathematical Logic 52 (7-8):779-791.
G-compactness and groups.Jakub Gismatullin & Ludomir Newelski - 2008 - Archive for Mathematical Logic 47 (5):479-501.
Lascar strong types in some simple theories.Steven Buechler - 1999 - Journal of Symbolic Logic 64 (2):817-824.
Lascar Strong Types in Some Simple Theories.Steven Buechler - 1999 - Journal of Symbolic Logic 64 (2):817-824.
| T|+‐resplendent models and the Lascar group.Enrique Casanovas & Rodrigo Peláez - 2005 - Mathematical Logic Quarterly 51 (6):626-631.
Galois groups as quotients of Polish groups.Krzysztof Krupiński & Tomasz Rzepecki - 2020 - Journal of Mathematical Logic 20 (3):2050018.
On two hierarchies of dimensions.Andreas Baudisch - 1987 - Journal of Symbolic Logic 52 (4):959-968.
On Superstable Expansions of Free Abelian Groups.Daniel Palacín & Rizos Sklinos - 2018 - Notre Dame Journal of Formal Logic 59 (2):157-169.

Analytics

Added to PP
2021-05-07

Downloads
11 (#1,164,628)

6 months
5 (#710,646)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Citations of this work

No citations found.

Add more citations

References found in this work

Galois groups of first order theories.E. Casanovas, D. Lascar, A. Pillay & M. Ziegler - 2001 - Journal of Mathematical Logic 1 (02):305-319.

Add more references