105 research outputs found
Quasiminimal structures, groups and Zariski-like geometries
We generalize Hrushovski's Group Configuration Theorem to quasiminimal
classes. As an application, we present Zariski-like structures, a
generalization of Zariski geometries, and show that a group can be found there
if the pregeometry obtained from the bounded closure operator is non-trivial
On model theory of covers of algebraically closed fields
We study covers of the multiplicative group of an algebraically closed field
as quasiminimal pregeometry structures and prove that they satisfy the axioms
for Zariski-like structures presented in \cite{lisuriart}, section 4. These
axioms are intended to generalize the concept of a Zariski geometry into a
non-elementary context. In the axiomatization, it is required that for a
structure \M, there is, for each , a collection of subsets of \M^n, that
we call the \emph{irreducible sets}, satisfying certain properties. These
conditions are generalizations of some qualities of irreducible closed sets in
the Zariski geometry context. They state that some basic properties of closed
sets (in the Zariski geometry context) are satisfied and that specializations
behave nicely enough. They also ensure that there are some traces of
Compactness even though we are working in a non-elementary context
Constructing strongly equivalent nonisomorphic models for unsuperstable theories, Part A
We study how equivalent nonisomorphic models an unsuperstable theory can
have. We measure the equivalence by Ehrenfeucht-Fraisse games
Constructing strongly equivalent nonisomorphic models for unsuperstable theories, Part C
In this paper we prove a strong nonstructure theorem for kappa (T)-saturated
models of a stable theory T with dop
Reduction of Database Independence to Dividing in Atomless Boolean Algebras
We prove that the form of conditional independence at play in database theory
and independence logic is reducible to the first-order dividing calculus in the
theory of atomless Boolean algebras. This establishes interesting connections
between independence in database theory and stochastic independence. As indeed,
in light of the aforementioned reduction and recent work of Ben-Yaacov [4], the
former case of independence can be seen as the discrete version of the latter
Borel* Sets in the Generalised Baire Space
We start by giving a survey to the theory of Borel*(\kappa) sets in the
generalized Baire space Baire({\kappa}) = {\kappa}^{\kappa}. In particular we
look at the relation of this complexity class to other complexity classes which
we denote by Borel({\kappa}), \Delta^1_1({\kappa}) and {\Sigma}^1_1({\kappa})
and the connections between Borel*(\kappa)-sets and the infinitely deep
language M_{{\kappa}^+{\kappa}}. In the end of the paper we prove the
consistency of Borel*(\kappa) \ne {\Sigma}^1_1({\kappa}).Comment: 19 page
On the reducibility of isomorphism relations
We study the Borel reducibility of isomorphism relations in the generalized
Baire space . In the main result we show for inaccessible
, that if is a classifiable theory and is stable with OCP,
then the isomorphism of models of is Borel reducible to the isomorphism of
models of
On the number of elementary submodels of an unsuperstable homogeneous structure
We show that if M is a stable unsuperstable homogeneous structure, then for
most kappa < |M|, the number of elementary submodels of M of power kappa is
2^kappa
An AEC framework for fields with commuting automorphisms
In this paper, we introduce an AEC framework for studying fields with
commuting automorphisms. Fields with commuting automorphisms generalise
difference fields. Whereas in a difference field, there is one distinguished
automorphism, a field with commuting automorphisms can have several of them,
and they are required to commute. Z. Chatzidakis and E. Hrushovski have studied
in depth the model theory of ACFA, the model companion of difference fields.
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)
On \Sigma^1_1-complete Equivalence Relations on the Generalized Baire Space
Working with uncountable structures of fixed cardinality, we investigate the
complexity of certain equivalence relations and show that if V = L, then many
of them are \Sigma^1_1-complete, in particular the isomorphism relation of
dense linear orders.
Then we show that it is undecidable in ZFC whether or not the isomorphism
relation of a certain well behaved theory (stable, NDOP, NOTOP) is
\Sigma^1_1-complete (it is, if V = L, but can be forced not to be).Comment: 22 page
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