We prove that a Kueker theory with infinite dcl(Ø) does not have the strict order property and that strongly minimal types are dense: any non-algebraic formula is contained in a strongly minimal type.

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 $\subset$ dcl(Q), such that M is prime over I. (shrink)

We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasi-endomorphisms of G must be an infinite finitely presented ring.

Non-$n$-ampleness as defined by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of $\Sigma$-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the early (...) stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable. (shrink)

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we (...) describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work. (shrink)

AssumeTis 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 typep, in case whenpis a complete type over a finite set of parameters. We show that for any flat Morley sequenceQthere is a modelMofTwhich isτ-atomic over {Q}. When additionallyThas few countable models and is 1-based, we prove that withinMthere is an infinite Morley sequenceI, withI⊂ dcl(Q), such thatMis prime overI.

Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.