Daniel Lascar

An introduction to forking

The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion. Consider some well-known examples of ℵ 0 -stable theories: vector spaces over Q , algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelahs book will convince the reader that this is the case. What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p .

Ranks and definability in superstable theories

We study the notion of definable type, and use it to define theproduct of types and theheir of a type. Then, in the case of stable and superstable theories, we make a general study of the notion of rank. Finally, we use these techniques to give a new proof of a theorem of Lachlan on the number of isomorphism types of countable models of a superstable theory.

Extending partial automorphisms and the profinite topology on free groups

A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1, p2, . . . , pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1, α2, . . . , αn of C3 extending the pi. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskĭı stating that a finite product of finitely generated subgroups is closed for this topology.

Hyperimaginaries and Automorphism Groups

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [?], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M). In section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In section ∗Partially supported by an NSF grant This work was begun during a visit of the two authors to the Centre de Recerca Matemèmatica, Institut d’Estudis Catalans. The authors wish to express their gratitude for its support and hospitability

Les beaux automorphismes

Assume that the class of partial automorphisms of the monster model of a complete theory has the amalgamation property. The beautiful automorphisms are the automorphisms of models ofT which: 1. are strong, i.e. leave the algebraic closure (inTeq) of the empty set pointwise fixed, 2. are obtained by the Fraïsse construction using the amalgamation property that we have just mentioned. We show that all the beautiful automorphisms have the same theory (in the language ofT plus one unary function symbol for the automorphism), and we study this theory. In particular, we examine the question of the saturation of the beautiful automorphisms. We also prove that in some cases (in particular if the theory is ω-stable andG-trivial), almost all (in the sense of Baire categoricity) automorphisms of the saturated countable model are beautiful and conjugate.

Les Automorphismes D'un Ensemble Fortement Minimal

Let 9 be a countable saturated structure, and assume that D(v) is a strongly minimal formula (without parameter) such that 9 is the algebraic closure of D(9). We will prove the two following theorems: THEOREM 1. If G is a subgroup of Aut(9) of countable index, there exists a finite set A in 9 such that every A-strong automorphism is in G. THEOREM 2. Assume that G is a normal subgroup of Aut(M) containing an element g such that for all n there exists X c D(9) such that Dim(g(X)/X) > n. Then every strong automorphism is in G. ?

Why Some People are Excited by Vaught's Conjecture

§I . In 1961, R. L. Vaught ([V]) asked if one could prove, without the continuum hypothesis, that there exists a countable complete theory with exactly ℵ 1 isomorphism types of countable models. The following statement is known as Vaught conjecture: Let T be a countable theory. If T has uncountably many countable models, then T has countable models . More than twenty years later, this question is still open. Many papers have been written on the question: see for example [HM], [M1], [M2] and [St]. In the opinion of many people, it is a major problem in model theory. Of course, I cannot say what Vaught had in mind when he asked the question. I just want to explain here what meaning I personally see to this problem. In particular, I will not speak about the topological Vaught conjecture, which is quite another issue. I suppose that the first question I shall have to face is the following: “Why on earth are you interested in the number of countable models—particularly since the whole question disappears if we assume the continuum hypothesis?” The answer is simply that I am not interested in the number of countable models, nor in the number of models in any cardinality, as a matter of fact. An explanation is due here; it will be a little technical and it will rest upon two names: Scott (sentences) and Morley (theorem).

Forking and fundamental order in simple theories

We give a characterisation of forking in the context of simple theories in terms of the fundamental order.

Omega-stable groups

This is a self-contained presentation of the theory of ω-stable groups. All proofs are given.