Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. LetKbe a regular field which is not generically stable and letpbe its global generic type. We observe that ifKhas a finite extensionLof degreen, thenPhas unbounded orbit under the action of (...) the multiplicative group ofL.Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable. (shrink)

We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of א₀-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (������, ≼ ������ ) be a simple finitary AEC, weakly categorical in some uncountable κ. Then (������, ≼ ������ ) is weakly categorical in (...) each λ ≥ min { \group{ \{\kappa,\beth_{ \group{ (2^{ \aleph_{ 0 _} ^});^{ + ^} \group} _}\}; \group} . If the class (������, ≼ ������ ) is also LS(������)-tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense. We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples. (shrink)

We study Lascar strong types and Galois types and especially their relation to notions of type which have finite character. We define a notion of a strong type with finite character, the so-called Lascar type. We show that this notion is stronger than Galois type over countable sets in simple and superstable finitary AECs. Furthermore, we give an example where the Galois type itself does not have finite character in such a class.

Quasiminimal structures play an important role in non-elementary categoricity. In this paper we explore possibilities of constructing quasiminimal models of a given first-order theory. We present several constructions with increasing control of the properties of the outcome using increasingly stronger assumptions on the theory. We also establish an upper bound on the Hanf number of the existence of arbitrarily large quasiminimal models.

We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...) as the independence relation itself, is uniquely determined. Finally, we show that an excellent class is simple if and only if it has extensible U-rank . We deduce that any excellent class of finite U-rank is simple, and that any uncountably categorical excellent class has an expansion with countably many constants which is simple. (shrink)

In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order non-forking over complete types. Our approach generalizes the context of 0-stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over 0/, Löwenheim–Skolem number being ω, -tameness and a property we call finite character. We also (...) start the studies of these classes from the 0-stable case. Stability in 0 and -tameness can be replaced by categoricity above the Hanf number. Finite character is the main novelty of this paper. Almost all examples of AEC’s have this property and it allows us to use weak types, as we call them, in place of Galois types. (shrink)

We show that any (atomic) excellent class K can be expanded with hyperimaginaries to form an (atomic) excellent class Keq which has canonical bases. When K is, in addition, of finite U-rank, then Keq is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class K gives rise to an excellent class, which (...) is simple if K is of finite U-rank. (shrink)