Enrique Casanovas

GALOIS GROUPS OF FIRST ORDER THEORIES

We study the groups GalL(T) and GalKP(T), and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP (a non G-compact theory). It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

On Elementary Equivalence for Equality-free Logic

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by provid- ing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fra¨´ type theorem. We also give characterizations of elementary classes in equality- free logic. As a by-product we characterize the sentences that are logically equivalent to an equality-free one. 1I ntroduction In first-order logic it is common to employ one symbol for the equality relation. Equality is considered a logical notion, with a fixed meaning. This was not the case when the first investigations in mathematical logic took place, but this practice has been strongly supported by successful applications to mathematical theories. Thus, the general study of first-order logic without equality, or equality-free logic ,a sw eprefer to call it, has been neglected in favor of the more powerful version with equality. Recently some interest in fragments of equality-free logic has arisen in the frame of algebraic logic (see Blok and Pigozzi (5) and Bloom (2)). We think that a model-theoretic study of equality-free logic is worthwhile by itself and we hope that, by means of contrast with the well-known results for first-order logic, this study will contribute to the understanding of the role of equality in mathematical theories and structures. As an easy example of this comparison consider the fact that every satisfiable set of equality-free sentences has an infinite model. Let L be a similarity type. The set of equality-free formulas of L, that is, the set of all first-order formulas of L not containing the equality symbol, is denoted by L − . Given two L-structures A, B with A ≡ − B we mean that A and B satisfy exactly the same sentences of L − .W edevote this paper to the study of algebraic characteri- zations of the relation ≡ − and of elementary classes in the sense of L − .

Stable Theories with a New Predicate

Let M be an L -structure and A be an infinite subset of M . Two structures can be defined from A : • The induced structure on A has a name R φ for every ∅-definable relation φ ( M ) ∩ A n on A . Its language is A with its L ind -structure will be denoted by A ind . • The pair ( M, A ) is an L(P) -structure, where P is a unary predicate for A and L(P) = L ∪{ P }. We call A small if there is a pair ( N, B ) elementarily equivalent to ( M, A ) and such that for every finite subset b of N every L –type over Bb is realized in N . A formula φ ( x, y ) has the finite cover property (f.c.p) in M if for all natural numbers k there is a set of φ –formulas which is k –consistent but not consistent in M. M has the f.c.p if some formula has the f.c.p in M . It is well known that unstable structures have the f.c.p. (see [6].) We will prove the following two theorems. Theorem A. Let A be a small subset of M. If M does not have the finite cover property then, for every λ ≥ ∣ L ∣, if both M and A ind are λ– stable then (M, A) is λ– stable . Corollary 1.1 (Poizat [5]). If M does not have the finite cover property and N ≺ M is a small elementary substructure, then (M, N) is stable . Corollary 1.2 (Zilber [7]). If U is the group of wots of unity in the field ℂ of complex numbers the pair (ℂ, U ) is ω – stable . Proof. (See [4].) As a strongly minimal set ℂ is ω–stable and does not have the f.c.p. By the subspace theorem of Schmidt [3] every algebraic set intersects U in a finite union of translates of subgroups definable in the group structure of U alone. Whence U ind is nothing more than a (divisible) abelian group, which is ω –stable.

The number of types in simple theories

Abstract We continue work of Shelah on the cardinality of families of pairwise incompatible types in simple theories obtaining characterizations of simple and supersimple theories. We develop a local analysis of the number of types in simple theories and we find a new example of a simple unstable theory.

Omitting Types in Incomplete Theories

We characterize omissibility of a type, or a family of types, in a countable theory in terms of non-existence of a certain tree of formulas. We extend results of L. Newelski on omitting K non-isolated types. As a consequence we prove that omissibility of a family of K types is equivalent to omissibility of each countable subfamily.

The free roots of the complete graph

There is a model-completion T n of the theory of a (reflexive) n-coloured graph such that R n is total, and R i o Rj ⊆ R i+j for all i,j. For n > 2, the theory T n is not simple, and does not have the strict order property. The theories T n combine to yield a non-simple theory Too without the strict order property, which does not eliminate hyperimaginaries.

Logical Operations and Invariance

I present a notion of invariance under arbitrary surjective mappings for operators on a relational finite type hierarchy generalizing the so-called Tarski–Sher criterion for logicality and I characterize the invariant operators as definable in a fragment of the first-order language. These results are compared with those obtained by Feferman and it is argued that further clarification of the notion of invariance is needed if one wants to use it to characterize logicality.

Weak forms of elimination of imaginaries

We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A B acl(A) and closed subgroups of the Galois group Aut(acl(A)/A). We also characterize when the topology of the Galois group is the quotient topology.

GENERIC STABILITY AND STABILITY

We prove two results about generically stable types p in arbitrary theories. The rst, on existence of strong germs, generalizes results from [3] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p (m) for all m. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If p(x) 2 S(B) is stable and does not fork over A then p A is stable. (They had solved some special cases.) 1 Stable and generically stable types Our notation is standard. We work with an arbitrary complete theory T in language L. C denotes a ‘monster model’. M denotes a small elementary submodel, and A;B;::: denote small subsets. L(C) denotes the collection of formulas with parameters from C, likewiseL(A) etc. We sometimes say A-invariant for Aut(C=A)-invariant. By a global type we mean a complete type over C. We assume familiarity with notions from model theory such as heir, coheir, denable type, forking. The book [7] is a good reference, but see also [5].

Normal hyperimaginaries

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of the Lascar-Pillay theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of the Peter–Weyl theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for bounded hyperimaginaries.