Ludomir Newelski

Small Profinite Groups

We propose a model-theoretic framework for investigating profinite groups. Within this framework we define and investigate small profinite groups. We consider the question if any small profinite group has an open abelian subgroup.

G-compactness and groups

Lascar described EKP as a composition of EL and the topological closure of EL (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/GL. We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories.

M-gap conjecture and m-normal theories

We find a small weakly minimal theory with an isolated weakly minimal type ofM-rank ∞ and an isolated weakly minimal type of arbitrarily large finiteM-rank. These examples lead to the notion of an m-normal theory. We prove theM-gap conjecture for m-normalT. In superstable theories with few countable models we characterize traces of complete types as traces of some formulas. We prove that a 1-based theory with few countable models is m-normal. We investigate generic subgroups of small superstable groups. We compare the notions of independence induced by measure (μ-independence) and category (m-independence).

On Bounded Type-Definable Equivalence Relations

We investigate some topological properties of the spaces of classes of bounded type-definable equivalence relations.

Scott analysis of pseudotypes

This is a continuation of [N2]. We find a Borel definition of Q -isolation. We pursue a topological and Scott analysis of pseudotypes on S(Q) .

TOPOLOGICAL DYNAMICS OF STABLE GROUPS

Assume G is a group definable in a model M of a stable theory T . We prove that the semigroup SG(M) of complete G-types over M is an inverse limit of some semigroups type-definable in M eq. We prove that the maximal subgroups of SG(M) are inverse limits of some definable quotients of subgroups of G. We consider the powers of types in the semigroup SG(M) and prove that in a way every type in SG(M) is pro-finitely many steps away from a type in a subgroup of SG(M).

Flat Morley sequences

Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p , in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ -atomic over { Q }. When additionally T has few countable models and is 1-based, we prove that within M there is an infinite Morley sequence I , with I ⊂ dcl( Q ), such that M is prime over I .

VAUGHT'S CONJECTURE FOR SOME MEAGER GROUPS

AssumeG is a superstable group ofM-rank 1 and the division ring of pseudo-endomorphisms ofG is a prime field. We prove a relative Vaught’s conjecture for Th(G). When additionallyU(G) =ω, this yields Vaught’s conjecture for Th(G).

Weakly minimal formulas: a global approach

Abstract Let T be a countable weakly minimal unidimensional theory and A a set of parameters. We determine the number of models of T(A) in power >¦A¦+ ʗ, depending on topological properties of the set of types realized in A.

Independence results for uncountable superstable theories

We prove that the following statement is independent of ZFC+┐CH: IFT is a superstable theory of power <2ℵ0,M≰N are models ofT withQ(M)=Q(N), then there isN′≱N withQ(N)=Q(N′). This generalizes Lachlan’s (1972) result.