We investigate some common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left$ofAof definable subgroups ofGover$acl\left$which are boolean combinations of (...) instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions. (shrink)

We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from 'primitive extensions' to the natural numbers a theory T μ of an expansion (...) of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable. (shrink)

We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.

Given a pseudovariety [MATHEMATICAL SCRIPT CAPITAL C], it is proved that a residually-[MATHEMATICAL SCRIPT CAPITAL C] superstable group G has a finite seriesG0 ⊴ G1 ⊴ · · · ⊴ Gn = Gsuch that G0 is solvable and each factor Gi +1/Gi is in [MATHEMATICAL SCRIPT CAPITAL C] . In particular, a residually finite superstable group is solvable-by-finite, and if it is ω -stable, then it is nilpotent-by-finite. Given a finitely generated group G, we show that if G is ω (...) -stable and satisfies some residual properties , then G is finite. (shrink)

Let $M$ be a saturated model of a superstable theory and let $G = \operatorname{Aut}(M)$. We study subgroups $H$ of $G$ which contain $G_{(A)}, A$ the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types $p$ in the context of $p$-simple types.

We prove that a nonabelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of nonabelian CSA-group of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of Mustafin and Poizat [E. Mustafin, B. Poizat, Sous-groupes superstables de SL2 ] which states that a superstable model of the (...) universal theory of nonabelian free groups is abelian. We deduce also that a superstable torsion-free hyperbolic group is cyclic. We close the paper by showing that an existentially closed CSA*-group is not superstable. (shrink)

We define an $\mathfrak{R}$-group to be a stable group with the property that a generic element can only be algebraic over a generic. We then derive some corollaries for $\mathfrak{R}$-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are $\mathfrak{R}$-groups.

We study and develop a notion of isogeny for superstable groups inspired by the notion in algebraic groups and differential algebraic notions developed by Cassidy and Singer. We prove several fundamental properties of the notion. Then we use it to formulate and prove a uniqueness results for a decomposition theorem about superstable groups similar to one proved by Baudisch. Connections to existing model theoretic notions and existing differential algebraic notions are explained.

In this paper, we shall survey results about the group-theoretic properties of stable groups. These can be classified into three main categories, according to the strength of the assumptions needed: chain conditions, generic types, and some form of rank. Each category has its typical application: Chain conditions often allow us to deduce global properties from local ones, generic properties are used to get definable groups from undefinable ones, and rank is necessary to interpret fields in certain group actions. While originally (...) the main input came from algebraic group theory, founded on the similarities between rank and dimension, increasingly methods borrowed from finite group theory have come to play a rôle, emphasizing the study of characteristic subgroups and characteristic families of subgroups. (shrink)

Let S and T be countable complete theories. We assume that T is superstable without the dimensional order property, and S is interpretable in T in such a way that every model of S is coded in a model of T. We show that S does not have the dimensional order property, and we discuss the question of whether $\operatorname{Depth}(S) \leq \operatorname{Depth}(T)$ . For Mekler's uniform interpretation of arbitrary theories S of finite similarity type into suitable theories T s of (...) groups we show that $\operatorname{Depth}(S) \leq \operatorname{Depth}(T_S) \leq 1 + \operatorname{Depth}(S)$. (shrink)

Ni konstruas korpojn de Morleja ranko du, kiuj estas ricevitaj per memsuficanta amalgameco de korpoj kun unara predikato, lau la Hrushovkija maniero.

Theorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let $A = \mathrm{acl}(\varnothing) \cap M$ and $I = \{r \in R: rM \subset A\}$ . Notice that I is an ideal. (i) F = R/I is a finite field. (ii) Suppose that a, b 0 ,...,b n ∈ M and a b̄. Then there are s, r i ∈ R, i ≤ n, such that sa + (...) ∑ i ≤ n r ib i ∈ A and $s \not\in I$ . It follows from Theorem A that algebraic closure in M is modular. Using this and results in [B1] and [B2], we obtain Theorem B. Let M be as in Theorem A. Then Vaught's conjecture holds for Th(M). (shrink)

A group is small if it has only countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite (...) weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary: a weakly small group with simple theory has an infinite definable finite-by-abelian subgroup. Secondly, in a group with simple theory, a solvable group A of derived length n is contained in an A-definable almost solvable group of class at most 2n — 1. (shrink)

A first order theory T of power λ is called unidimensional if any twoλ+-saturated models of T of the same cardinality are isomorphic. We prove here that such theories are superstable, solving a problem of Shelah. The proof involves an existence theorem and a definability theorem for definable groups in stable theories, and an analysis of their relation to regular types.

We prove that a nonabelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of nonabelian CSA-group of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of Mustafin and Poizat [E. Mustafin, B. Poizat, Sous-groupes superstables de SL2 ] which states that a superstable model of the (...) universal theory of nonabelian free groups is abelian. We deduce also that a superstable torsion-free hyperbolic group is cyclic. We close the paper by showing that an existentially closed CSA*-group is not superstable. (shrink)

First, it is pointed out how the author's new differential Galois theory contributes to the understanding of the differential closure of an arbitrary differential field . Secondly, it is shown that a superstable differential field has no proper differential Galois extensions.

A first order theory T of power λ is called unidimensional if any twoλ+-saturated models of T of the same cardinality are isomorphic. We prove here that such theories are superstable, solving a problem of Shelah. The proof involves an existence theorem and a definability theorem for definable groups in stable theories, and an analysis of their relation to regular types.