arXiv subjects

Kaisa Kangas

Publications and source records attributed to Kaisa Kangas.

7 recordsLinked to original sources

arXiv2022

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 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).

math.LO↗

arXiv2018

Categoricity and Universal Classes

Let $(\mathcal{K} ,\subseteq )$ be a universal class with $LS(\mathcal{K})=λ$ categorical in regular $κ>λ^+$ with arbitrarily large models, and let $\mathcal{K}^*$ be the class of all $\mathcal{A}\in\mathcal{K}_{>λ}$ for which there is $\mathcal{B} \in \mathcal{K}_{\geκ}$ such that $\mathcal{A}\subseteq\mathcal{B}$. We prove that $\mathcal{K}^*$ is categorical in every $ξ>λ^+$, $\mathcal{K}_{\ge\beth_{(2^{λ^+})^+}} \subseteq \mathcal{K}^{*}$, and the models of $\mathcal{K}^*_{>λ^+}$ are essentially vector spaces (or trivial i.e. disintegrated).

math.LO↗

arXiv2017

Finding a field in a Zariski-like structure

We show that if $\M$ is a Zariski-like structure (see \cite{lisuriart}) that does not interpret a non-classical group, and the canonical pregeometry obtained from the bounded closure operator (bcl) is non locally modular, then $\M$ interprets an algebraically closed field.

math.LO↗

arXiv2017

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.

math.LO↗

arXiv2015

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 $n$, 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.

math.LO↗

arXiv2014

Finding groups in Zariski-like structures

We study quasiminimal classes, i.e. abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. For these classes, we develop an independence notion, and in particular, a theory of independence in $\M^{eq}$. We then generalize Hrushovski's Group Configuration Theorem to our setting. In an attempt to generalize Zariski geometries to the context of quasiminimal classes, we give the axiomatization for Zariski-like structures, and as an application of our group configuration theorem, show that groups can be found in them assuming that the pregeometry obtained from the bounded closure operator is non-trivial. Finally, we study the cover of the multiplicative group of an algebraically closed field and show that it provides an example of a Zariski-like structure.

math.LO↗

arXiv2012

On Second-order Characterizability

We investigate the extent of second order characterizable structures by extending Shelah's Main Gap dichotomy to second order logic. For this end we consider a countable complete first order theory T. We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second order logic with Henkin semantics.

math.LO↗