The strict order property and generic automorphisms

The small index property of automorphism groups of ab-initio generic structures

Suppose $M$ is a countable ab-initio (uncollapsed) generic structure which is obtained from a pre-dimension function with rational coefficients. We show that if $H$ is a subgroup of $\mbox{Aut}\left(M\right)$ with $\left[\mbox{Aut}\left(M\right):H\right]<2^{\aleph_{0}}$, then there exists a finite set $A\subseteq M$ such that $\mbox{Aut}_{A}\left(M\right)\subseteq H$. This shows that $\mbox{Aut}\left(M\right)$ has the small index property.

Model companion of ordered theories with an automorphism

Kikyo and Shelah showed that if $T$ is a theory with the Strict Order Property in some first-order language $\mathcal{L}$, then in the expanded language $\mathcal{L}_σ:= \mathcal{L}\cup\{σ\}$ with a new unary function symbol $σ$, the bigger theory $T_σ:= T\cup\{``σ\mbox{is an} \mathcal{L}\mbox{-automorphism''}\}$ does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory $T_σ$ as the base theory $T$ together with a ``restricted'' class of automorphisms, then $T_σ$ can have a model companion in $\mathcal{L}_σ$. We show this in t…

The structure of random automorphisms of countable structures

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different …

$n$-部竞赛图与半通用竞赛图的部分自同构的扩张性质

Extension property for partial automorphisms of the $n$-partite and semigeneric tournaments

本文证明有限 $n$-部竞赛图（$n$ 从2到$\omega$）与有限半通用竞赛图均满足EPPA，并证明相应的通用结构具有充足泛型。

Hall universal group has ample generic automorphisms

We show that the automorphism group of Philip Hall's universal locally finite group has ample generics,that is, it admits comeager diagonal conjugacy classes in all dimensions.Consequently, it has the small index property, is not the union of a countable chain of non-open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall universal group. For example, we show that every generic automorphism of Hall universal group is conjugate to all of its powers, and hence has roots of all orders.

Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures

We define a simple criterion for a homogeneous, complete metric structure $X$ that implies that the automorphism group $\mbox{Aut}(X)$ satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space $\mbox{U}$, the Lebesgue probability measure algebra $\mbox{MALG}$, and the Hilbert space $\ell_2$, thus proving that $\mbox{Iso}(\mbox{U})$, $\mbox{Aut}(\mbox{MALG})$, $U(\ell_2)$, and $O(\ell_2)$ share these properties. We also form…

On some dynamical aspects of NIP theories

We study some dynamical aspects of the action of automorphisms in model theory in particular in the presence of invariant measures. We give some characterizations for NIP theories in terms of dynamics of automorphisms and invariant measures for example in terms of compact systems, entropy and measure algebras. Moreover, we study the concept of symbolic representation for models. Amongst the results, we give some characterizations for dividing lines and combinatorial configurations such as independence property, order property and strictly order property in terms of symbolic representations.

赋值Hahn群的自同构

Automorphisms of valued Hahn groups

本文研究了满足特定提升性质的Hahn群的自同构结构，通过半直积分解定理和矩阵描述揭示了其自同构群的性质。

Turbulence, amalgamation and generic automorphisms of homogeneous structures

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which…

一个参数化的$\diamondsuit$用于Laver性质和$\mathcal P(ω)/\mathrm{Fin}$的非平凡自同构

A parametrized $\diamondsuit$ for the Laver property and nontrivial automorphisms of $\mathcal P(ω)/\mathrm{Fin}$

本文提出参数化$\diamondsuit(\mathsf{LP})$原理，证明其在特定forcing模型中成立，并展示其导致$\mathcal P(ω)/\mathrm{Fin}$存在非平凡自同构。

Continuous logic and the strict order property

We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.