Let T be a countable, complete, ω-stable, nonmultidimensional theory. By Lascar [7], in T eq there is in every dimension of T a type with Lascar rank ω α for some α. We give sufficient conditions for α to coincide with the level of that dimension in Pillay's [10] RK-hierarchy of dimensions computed in T eq . In particular, this is fulfilled for modules.

This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for L ω 1 ,ω (Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) (...) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry. (shrink)

In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma /\Delta $ for each definable convex subgroup $\Delta $, and sorts for the quotient groups $\Gamma /(\Delta + \ell \Gamma )$ where $\Delta $ is a definable convex subgroup and $\ell \in \mathbb {N}_{\geq 2}$. (...) We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\ell \Gamma $ in $\Gamma $, where $\ell \in \mathbb {N}_{\geq 2}$. (shrink)

We investigate some relations between omitting types of a countable theory and some notions defined in terms of the real line, such as for example the ideal of meager subsets ofR. We also try to express connections between the logical structure of a theory and the existence of its countable models omitting certain families of types.It is well known that assuming MA we can omit (...) able to omit shrink)

We present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B |<κ(T ) pieces, Ai | i < |B |<κ(T ) , such that for each Ai there is a Bi ⊆ B where |Bi| < κ(T ) and Ai..

The paper continues [1]. Let S be a complete theory of ultraflat (e.g. planar) graphs as introduced in [4]. We show a strong form of NOTOP for S: The union of two models M1 and M2, independent over a common elementary submodel M0, is the primary model over M1 ∪ M2 of S. Then by results of [1] Mekler's construction [6] gives for such a theory S of nice ultraflat graphs a superstable 2-step-nilpotent group of exponent $p (>2)$ with NDOP (...) and NOTOP. (shrink)

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have (...) several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). (shrink)

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.

LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G) (...) questions, and improves results, of Faber, Laver and McKenzie. (shrink)

We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe [Formula: see text] which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided [Formula: see text] and λ ≥ χ. For the missing case when [Formula: see text], we prove that [Formula: see text] is totally categorical provided that [Formula: see text] is categorical in [Formula: see text] and [Formula: see text].

The role played by real-valued functions in functional analysis is fundamental. One often considers metrics, or seminorms, or linear functionals, to mention some important examples. We introduce the notion of definable real-valued function in functional analysis: a real-valued function f defined on a structure of functional analysis is definable if it can be "approximated" by formulas which do not involve f. We characterize definability of real-valued functions in terms of a purely topological condition which does not involve logic.

We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ≤ 2ℵ0 associated with nonprincipal ultrafilters on ℕ. They are either all isomorphic, or else there are 22ℵ0 many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II1 factors, as well as their relative commutants and include several applications. We also show that the CAF001-algebra [Formula: see text] always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters (...) on ℕ. (shrink)

he model theory of groups of unitriangular matrices over rings is studied. An important tool in these studies is a new notion of a quasiunitriangular group. The models of the theory of all unitriangular groups are algebraically characterized; it turns out that all they are quasiunitriangular groups. It is proved that if R and S are domains or commutative associative rings then two quasiunitriangular groups over R and S are isomorphic only if R and S are isomorphic or antiisomorphic. This (...) algebraic result is new even for ordinary unitriangular groups. The groups elementarily equivalent to a single unitriangular group UTn are studied. If R is a skew field, they are of the form UTn, for some S ≡ R. In general, the situation is not so nice. Examples are constructed demonstrating that such a group need not be a unitriangular group over some ring; moreover, there are rings P and R such that UTn ≡ UTn, but UTn cannot be represented in the form UTn for S ≡ R. We also study the number of models in a power of the theory of a unitriangular group. In particular, we prove that, for any communicative associative ring R and any infinite power λ, I = I). We construct an associative ring such that I = 3 and I) = 2. We also study models of the theory of UTn in the case of categorical R. (shrink)

Let L(Q) be first order logic with Keisler's quantifier, in the λ + interpretation (= the satisfaction is defined as follows: $M \models (\mathbf{Q}x)\varphi(x)$ means there are λ + many elements in M satisfying the formula φ(x)). Theorem 1. Let λ be a singular cardinal; assume □ λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits $p (= p$ has no support, to (...) be defined in $\S1$ ), then T has a model of cardinality λ + in the λ + interpretation which omits p. Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □ λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ + such that a formula φ(x) is realized by λ + elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming $\lambda = \lambda^{ . It is new when λ is singular or when |T| = λ is regular. Theorem 3. Let λ be singular. If $\operatorname{Con}(ZFC + GCH + (\exists\kappa)$ [κ is a strongly compact cardinal]), then the following in consistent: ZFC + GCH + the conclusions of all above theorems are false. (shrink)

We introduce a new device in the study of abstract elementary classes : Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. We show:Theorem 0.1 An AEC K is fully \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa = \beth _{\kappa } > \text {LS}$$\end{document}. If K is Galois stable, then the (...) class of κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Galois saturated models of K admits an independence notion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory. (shrink)

We investigate the notion of a semi-retraction between two first-order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.

We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in (...) these PAC substructures and show several results related to the SOPnhierarchy. Fifth, we characterize the algebraic closure in PAC structures. (shrink)

This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD(A), where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an∈ P and finite subset C ⊆ Q, but (...) [Formula: see text] for some independent a1, …, an, an+1∈ P and some finite subset C ⊆ Q.Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] interprets a non-classical group, or n = 1,2,3 and•If n = 1 then G is abelian and acts regularly on P′.•If n = 2 the action of G on P′ is isomorphic to the affine action of K ⋊ K* on the algebraically closed field K.•If n = 3 the action of G on P′ is isomorphic to the action of PGL2(K) on the projective line ℙ1(K) of the algebraically closed field K.We prove a similar result for excellent classes. (shrink)

There are shown to many ε-Hermitian spaces, and an isometry criterion is stated which holds under MA ℵ 1 and is false under $2^{\aleph_0}.

This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of \ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete...

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank (...) function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… ungroupede Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:Theorem 1.1.A ranked structure is superstable.An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5. (shrink)

We try to count the number of countable models M of T with a fixed set Q = φ (M) of realizations of a type φ. Also, for stable T, we define an ordinal rank M measuring multiplicity of types, with additivity properties similar to those of U-rank.

Let T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2 λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least $\min(2^\lambda,\beth_2)$ resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, (...) 2 ω } there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ. (shrink)

We prove two results on the stability spectrum for Lω1,ω. Here [Formula: see text] denotes an appropriate notion of Stone space of m-types over M. Theorem for unstable case: Suppose that for some positive integer m and for every α μ, K is not i-stable in μ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness (...) in the theory under study. In this paper, we expound the construction of tree indiscernibles for sentences of Lω1,ω. Further we provide some context for a number of variants on the Ehrenfeucht–Mostowski construction. (shrink)

We give here alternative definitions for the notions that S. Shelah has introduced in recent papers: the dimensional order property and the depth of a theory. We will also give a proof that the depth of a countable theory, when defined, is an ordinal recursive in T.