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Alf Onshuus

Publications and source records attributed to Alf Onshuus.

At least 19 recordsLinked to original sources
arXiv2025

Limit Semigroups with 2 Generators

In this paper we study the limit theory of numerical semigroups with two generators. We give a complete axiomatization in some cases.

math.LO↗
arXiv2025

Definably amenable groups in Continuous logic

We introduce the notions of definable amenability and extreme definable amenability for groups in continuous structures and conduct an extensive analysis of them, drawing parallels with the classical first-order case. We characterize both notions using fixed-point properties. We show that stable and ultracompact groups are definably amenable and prove that, for groups definable in dependent theories, definable amenability is equivalent to the existence of an f-generic type. Finally, we show the randomizations of first-order definably amenable groups are extremely definably amenable.

math.LO↗
arXiv2025

Continuous isomorphisms between groups definable in o-minimal expansions of the real field

In this paper we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field, which we will refer to as ``definable groups''. With this terminology, it is known (\cite{Pi88}) that any definable group is a Lie group, and in \cite{COP} a complete characterization of when a Lie group is \emph{Lie isomorphic} to a definable group'' was given. We continue the analysis by explaining when a Lie isomorphism between definable groups is definable. Among other things, we generalize Wilkie's result on the o-minimality of the exponential function (\cite{Wilkie}) by completely characterizing when, given an o-minimal expansion $\mathcal R$ of the real field and a Lie isomorphisms $ϕ$ between two $\mathcal R$-definable groups $G_1, G_2$, $ϕ$ can be added to the language of $\mathcal R$ preserving o-minimality. We also prove that any definable group $G$ can be endowed with an analytic manifold structure definable in $\mathcal R_{\text{Pfaff}}$ that makes it an analytic group.

math.LO↗
arXiv2021

Dependent finitely homogneneous rosy structures

We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for $ω$-stable $ω$-categorical structures. In particular, we prove that such structures have finite þ-rank and are coordinatized by a þ-rank 1 set. We show that they admit a distal, finitely axiomatizable, expansion. These results show that there are, up to inter-definability, at most countably many dependent rosy structures M which are homogeneous in a finite relational language.

math.LO↗
arXiv2020

A definability criterion for connected Lie groups

It has been known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in \cite{COSsolvable} in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group $G$ is Lie isomorphic to a group definable in an $o$-minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately we cannot achieve a full characterization. Assuming that a Lie group $G$ has a "good Levi descomposition", we prove that in order for $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in \cite{COSsolvable}.

math.LO↗
arXiv2018

Uniform Continuity in Definable Groups

In this paper we study analogues of amenability for topological groups in the context of definable structures. We prove fixed point theorems for such groups. More importantly, we propose definitions for definable actions and continuous functions from definable groups to topological spaces which might prove useful in other contexts.

math.LO↗
arXiv2018

Definable groups in models of Presburger Arithmetic and G^{00}

This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z, +)^{n} mod out by a lattice.

math.LO↗
arXiv2018

Stabilizers, groups with f-generics in NTP2 and PRC fields

In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably amenable NIP groups to NTP2 theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded PRC fields.

math.LO↗
arXiv2015

Metric Entropy estimation using o-minimality Theory

It is shown how tools from the area of Model Theory, specifically from the Theory of o-minimality, can be used to prove that a class of functions is VC-subgraph (in the sense of Dudley, 1987), and therefore satisfies a uniform polynomial metric entropy bound. We give examples where the use of these methods significantly improves the existing metric entropy bounds. The methods proposed here can be applied to finite dimensional parametric families of functions without the need for the parameters to live in a compact set, as is sometimes required in theorems that produce similar entropy bounds (for instance Theorem 19.7 of van der Vaart, 1998).

math.ST↗
arXiv2015

Groups in NTP2

We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.

math.LO↗
arXiv2014

Decomposable ordered groups

Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.

math.LO↗
arXiv2011

Additivity of the dp-rank

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.

math.LO↗
arXiv2011

The algebraic numbers definable in various exponential fields

We prove the following theorems: Theorem 1: For any E-field with cyclic kernel, in particular $\mathbb C$ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: For the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.

math.LO↗
arXiv2009

Orthogonality and domination in unstable theories

In the first part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. Then we try to develop analogous theory for arbitrary dependent theories.

math.LO↗