Superstable Theory Research Articles

Some semilattices of definable sets in continuous logic

In continuous first-order logic, the union of definable sets is definable but generally the intersection is not. This means that in any continuous theory, the collection of $\varnothing$-definable sets in one variable forms a join-semilattice under inclusion that may fail to be a lattice. We investigate the question of which semilattices arise as the collection of definable sets in a continuous theory. We show that for any non-trivial finite semilattice $L$ (or, equivalently, any finite lattice $L$), there is a superstable theory $T$ whose semilattice of definable sets is $L$. We then extend this construction to some infinite semilattices. In particular, we show that the following semilattices arise in continuous theories: $\alpha+1$ and $(\alpha+1)^\ast$ for any ordinal $\alpha$, a semilattice containing an exact pair above $\omega$, and the lattice of filters in $L$ for any countable meet-semilattice $L$. By previous work of the author, this establishes that these semilattices arise in stable theories. The first two are done in languages of cardinality $\aleph_0 + |\alpha|$, and the latter two in countable languages.

The isomorphism relation of theories with S-DOP in the generalised Baire spaces

We study the Borel-reducibility of isomorphism relations in the generalised Baire space κκ. In the main result we show for inaccessible κ, that if T is a classifiable theory and T′ is superstable with the strong dimensional order property (S-DOP), then the isomorphism of models of T is Borel reducible to the isomorphism of models of T′. In fact we show the consistency of the following: If κ is inaccessible and T is a superstable theory with S-DOP, then the isomorphism of models of T is Σ11-complete.

Strict Superstablity and Decidability of Certain Generic Graphs

We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ \alpha\in\omega+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $ \alpha. $ Furthermore, we show that these theories are decidable and pseudofinite.

\mathbf P-NDOP and \mathbf P-decompositions of ℵε-saturated models of superstable theories

Given a complete, superstable theory, we distinguish a class P of regular types, typically closed under automorphisms of C and non

Model theory of a Hilbert space expanded with an unbounded closed selfadjoint operator

We study a closed unbounded self-adjoint operator Q acting on a Hilbert space H in the framework of Metric Abstract Elementary Classes (MAECs). We build a suitable MAEC for such a structure, prove it is ℵ0-categorical and ℵ0-stable up to a system of perturbations. We give an explicit continuous axiomatization for the class. We also characterize non-splitting and show it has the same properties as non-forking in superstable first order theories. Finally, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.

The Schröder-Bernstein property for 𝑎-saturated models

A first-order theoryTThas theSchröder-Bernstein (SB) propertyif any pair of elementarily bi-embeddable models are isomorphic. We prove thatTThas an expansion by constants with the SB property if and only ifTTis superstable and non-multidimensional. We also prove that among superstable theoriesTT, the class ofaa-saturated models ofTThas the SB property if and only ifTThas no nomadic types.

Simple groups and the number of countable models

Let $T$ be a complete, superstable theory with fewer than $2^{\aleph_{0}}$ countable models. Assuming that generic types of infinite, simple groups definable in $T^{eq}$ are sufficiently non-isolated we prove that $\omega^{\omega}$ is the strict upper bound for the Lascar rank of $T$.

Rank and Dimension in Difference-Differential Fields

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model of DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent.

Preservation and violation of homogeneity of models under a special expansion

Conditions are found under which homogeneity of a model M implies homogeneity of the modelM eq. One of these conditions is ω-stability of the theory of M. Examples are constructed of superstable theories for which homogeneity is not preserved under such a passage.

On homogeneity and small extensions of models of the theory of linear ordering

Some properties concerning homogeneity and the existence of small extensions proved earlier for models of superstable theories, are proved here for models of the theory of linear ordering.

Vaught’s conjecture for superstable theories of finite rank

In [R. Vaught, Denumerable models of complete theories, in: Infinitistic Methods, Pregamon, London, 1961, pp. 303–321] Vaught conjectured that a countable first order theory has countably many or 2 ℵ 0 many countable models. Here, the following special case is proved. Theorem If T is a superstable theory of finite rank with < 2 ℵ 0 many countable models, then T has countably many countable models. The basic idea is to associate with a theory a ⋀ -definable group G (called the structure group) which controls the isomorphism types of countable models of the theory. The theory of modules is used to show that for M ⊧ T , G ∩ M is, essentially, the direct sum of copies of finitely many finitely generated subgroups. This is the principal ingredient in the proof of the following main theorem, from which Vaught’s conjecture follows immediately. Structure Theorem Let T be a countable superstable theory of finite rank with < 2 ℵ 0 many countable models. Then for M a countable model of T there is a finite A ⊂ M and a J ⊂ M such that M is prime over A ∪ J , J is A -independent and { stp ( a / A ) : a ∈ J } is finite.

Model-theoretic properties of regular polygons

This work is devoted to results obtained in the model theory of regular polygons. We give a characterization of monoids with axiomatizable and model-complete class of regular polygons. We describe monoids with complete class of regular polygons that satisfy some additional conditions. We study monoids whose regular core is represented as a union of finitely many principal right ideals and all regular polygons over which have a stable and superstable theory. We prove the stability of the class of all regular polygons over a monoid provided this class is axiomatizable and model-complete. We also describe monoids for which the class of all regular polygons is superstable and ω-stable provided this class is axiomatizable and model-complete.

Non-isolated types in stable theories

We introduce notions of strong and eventual strong non-isolation for types in countable, stable theories. For T superstable or small stable we prove a dichotomy theorem: a regular type over a finite domain is either eventually strongly non-isolated or is non-orthogonal to a NENI type (in T e q ). As an application we obtain the upper bound for Lascar’s rank of a superstable theory which is one-based or trivial, and has fewer than 2 ℵ 0 non-isomorphic countable models.

Invariant Version of Cardinality Quantifiers in Superstable Theories

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula.

Classifiable theories without finitary invariants

It follows directly from Shelah’s structure theory that if T is a classifiable theory, then the isomorphism type of any model of T is determined by the theory of that model in the language L ∞ , ω 1 ( d . q . ) . Leo Harrington asked if one could improve this to the logic L ∞ , ℵ ϵ ( d . q . ) In [S. Shelah, Characterizing an ℵ ϵ -saturated model of superstable NDOP theories by its L ∞ , ℵ ϵ -theory, Israel Journal of Mathematics 140 (2004) 61–111] Shelah gives a partial positive answer, showing that for T a countable superstable NDOP theory, two ℵ ϵ -saturated models of T are isomorphic if and only if they have the same L ∞ , ℵ ϵ ( d . q ) -theory. We give here a negative answer to the general question by constructing two classifiable theories, each with 2 ℵ 1 pairwise non-isomorphic models of cardinality ℵ 1 , which are all L ∞ , ℵ ϵ ( d . q . ) -equivalent, a shallow depth 3 ω -stable theory and a shallow NOTOP depth 1 superstable theory. In the other direction, we show that in the case of an ω -stable depth 2 theory, the L ∞ , ℵ ϵ ( d . q ) -theory is enough to describe the isomorphism type of all models.

Characterizing an ℵ ∈ -saturated model of superstable NDOP theories by its $$\mathbb{L}_{\infty ,\aleph _ \in ^ - } $$ -theory

Assume a complete countable first order theory is superstable with NDOP. We know that any ℵ ∈ -saturated model of the theory is ℵ ∈ -prime over a non-forking tree of “small” models and its isomorphism type can be characterized by its\(\mathbb{L}_{\infty ,k} \) (dimension qualifiers)-theory, or, if you prefer, appropriate cardinal invariants. We go one step further by providing cardinal invariants which are as finitary as seem reasonable.

On model complete differential fields

We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODEXX. We show that this sub-family is usually definable (in particular ifXXlives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.

On the Number of Countable Models of a Countable Supersimple Theory

It is proved that the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan's theorem on a superstable theory.

Geometry of *-Finite Types

AbstractAssumeTis a superstable theory with &lt; 2ℵ0countable models. We prove that any *- algebraic type of-rank &gt; 0 is m-nonorthogonal to a *-algebraic type of-rank 1. We study the geometry induced by m-dependence on a *-algebraic typep*of-rank 1. We prove that after some localization this geometry becomes projective over a division ring. Associated withp*is a meager typep. We prove thatpis determined byp*up to nonorthogonality and thatunderlies also the geometry induced by forking dependence on any stationarization ofp. Also we study some *-algebraic *-groups of-rank 1 and prove that any *-algebraic *-group of-rank 1 is abelian-by-finite.

Flat Morley sequences

AbstractAssumeTis 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.