Abstract
AbstractIn 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 $\operatorname {stp}(ac)$ is abelian, and the underlying theory is G-compact, then the Lascar groups of $\operatorname {stp}(ac)$ and of $\operatorname {stp}(a)$ 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 $\operatorname {stp}(ac)$ is abelian, nor the assumption of c being finite can be removed.
Highlights
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, the Lascar groups of stp(ac) and of stp(a) are isomorphic
We prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup
Summary
Note that if T is simple since T is G-compact, each GalλL(p) is a compact (i.e., quasi-compact and Hausdorff) connected group, so it is divisible (see [5, Theorem 9.35]). (1) We say p has abelian (or commutative) amalgamation of r and s, if there are independent a, b, c, d |= p such that ab, cd |= r and ac, bd |= s. As before it is enough to prove that a ≡L b It follows from the extension axiom for Lascar types together with Fact 0.14(1)(b), there are b′ ≡L b and a finite independent sequence (di)0≤i≤2n+2 of realizations of p satisfying the following conditions:. The answer to this question is yes if any two Lascar equivalence classes in p are interdefinable, since essentially this property (Remark 0.15(1)) implied the results when p is the type of a model. If Gal1L(p) is abelian any two Lascar equivalence classes in p are interdefinable: Let a, b |= p, and f ∈ Aut(p). A type of a model does not have reversible amalgamation either
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