Saharon Shelah
Solecki proved that the group of automorphisms of a countable structure cannot be an uncountable free abelian group. See more in Just, Shelah and Thomas math.LO/0003120 where as a by product we can say something on on uncountable structures. We prove here the following Theorem: If A is a countable model, then Aut(M) cannot be a free uncountable group.
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We prove that if the bounding number (d) is bigger than the reaping number (r), then the latter has uncountable cofinality.
In this paper, we compute universal minimal flows of groups of automorphisms of uncountable $ω$-homogeneous graphs, $K_n$-free graphs, hypergraphs, partially ordered sets, and their extensions with an $ω$-homogeneous ordering. We present an easy construction of such structures, expanding the jungle of extremely amenable groups.
D'Aquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $ω_1$-like models of PA.
Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality κ. We denote by μ(respectively \hatμ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. We prove that I(T,κ), as a function of κ> \aleph_0, is the minimum of 2^κ and one of the following functions: 1. 2^κ; 2. the constant function 1; 3. |\hatμ^n/{\sim_G}|-|(\hatμ- 1)^n/{\sim_G}| if \hatμ<ωfor some 1<n<ωand \hatμif \hatμ>= ωsome group G <= Sym(n); 4. the constant function \beth_2; 5. \be…
Fixed Sets of Automorphisms of Countable, Arithmetically Saturated Structures
研究可数算术饱和结构自同构的固定集的特征，采用算术饱和性理论和模型论方法，揭示自同构在结构中的固定行为规律
Every mathematical structure has an elementary extension to a pseudo-countable structure, one that is seen as countable inside a suitable class model of set theory, even though it may actually be uncountable. This observation, proved easily with the Boolean ultrapower theorem, enables a sweeping generalization of results concerning countable models to a rich realm of uncountable models. The Barwise extension theorem, for example, holds amongst the pseudo-countable models -- every pseudo-countable model of ZF admits an end extension to a model of ZFC+V=L. Indeed, the class of pseudo-countable m…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These counterexamples are often crucial in establishing reversals in the Reverse Mathematics program. All the previous is essentially limited to a language that can only express countable mathematics directly. The aim of this paper is to show that reversals and recursive counterexamples, countable in nature as they might be, directly yield new and interesting results abo…
We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.
In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density $κ$, for uncountable $κ$ such that $κ^{<κ}=κ$, is not a universal topological group of weight $κ$. We investigate automorphism groups of other uncountable ultrahomogeneous structures and prove that they are rarely universal topological groups for the corresponding classes. Our list of uncountable ultrahomogeneous structures includes random uncountable graph, tournament, linear order, partial order, group. That is in contrast with similar results obtained for automorphism groups of c…
For every countable structure $M$ we construct an $\aleph_0$-stable countable structure $N$ such that $Aut(M)$ and $Aut(N)$ are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure $M$ from the topological properties of the Polish group $Aut(M)$.
If an automorphism f of a structure M is such that fix(f^k) = fix(f) for all positive k, then M|fix(f) is a substructure of M. The possible isomorphism types of such M|fix(f) are characterized when M is countable and arithmetically saturated.
We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0 < k < ω$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < ω$ and $x^k = y$, then $x = e$. In particular, the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes results from [3] and [5], where this is proved for free and free Abelian uncountable groups.
If M,N are countable, arithmetically saturated models of Peano Arithmetic and Aut(M) is isomorphic to Aut(N), then the Turing-jumps of Th(M) and Th(N) are recursively equivalent.
We extend a dichotomy between 1-basedness and supersimplicity proved in a previous paper. The generalization we get is to arbitrary language, with no restrictions on the topology (we do not demand type-definabilty of the open set in the definition of essential 1-basedness). We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable.
We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is that the natural generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals.
Answering a question of Juhasz, Soukup and Szentmiklóssy we show that it is consistent that some first countable space of uncountable weight does not contain an uncountable subspace which has an irreducible base.
We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an uncountable set of countable models that are pairwise separable, then actually it has such a set of size $2^{\aleph_0}$. Our result follows trivially assuming the Continuum Hypothesis ($CH$). We work here in $ZFC$ (only without $CH$).
We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $κ$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $κ$, (2) $|\bar{a}\cap T|=κ$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $κ$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{ω,κ\}$.
We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that 1_P forces that ``X is a countable union of 0-dimensional subspaces of countable weight.'' We also show that this theorem is sharp in two different senses: (i) we can't get rid of using generic extensions, (ii) we have to consider all finite powers of $X$.
Kaplansky decompositions of Polish modules
本文刻画了 admits Kaplansky 分解的不可数 Polish 模，并利用该刻画推广了 Shelah 和 Solecki 的结果，给出了 admits 自由不可数 Polish 模的可数环的纯环论描述。
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 …
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdös--Rényi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measure…
The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts
本文研究可数递归饱和皮亚诺算术模型的自同构群子群，证明其小索引性质、余合集为不可数集，并指出非平凡正规子群在其中为渺小集，特别说明无限循环群不是其同态像。
We will review the main results concerning the automorphism groups of saturated structures which were obtained during the two last decades. The main themes are: the small index property in the countable and uncountable cases; the possibility of recovering a structure or a significant part of it from its automorphism group; the subgroup of strong automorphisms.
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.
Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product (X_omega Z) to G, then there exists a finite subset F of omega and a homomorphism h:*_{i in F} Z --> G such that h=h rho_F, where rho_F is the natural map from (X_{i in omega})Z to *_{i in F}Z . Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman's result fails if one passes from countable to uncountable. In particular, we show that fo…
Uncountable homogeneous structures
本文研究具有给定年龄的不可数齐次一阶结构是否存在，并通过可数齐次结构的自嵌入幺半群及其合并性质，结合抽象 Fraïssé 理论给出构造方法。
Generalized Descriptive Set Theory and Classification Theory
本文研究了不可数情况下的广义描述集论，探讨了等价关系的伯恩-可约性，并揭示了第一阶理论稳定性复杂性与同构关系复杂性之间的联系。
Combinatorial covering properties in an uncountable setting: canonical examples
本文通过推广经典可数构造，在不可数设定下构造了满足Hurewicz、Menger和γ-性质的空间例子，并揭示了多个新开问题。
What would the rational Urysohn space and the random graph look like if they were uncountable?
本文推广递增实数集概念以建立不可数结构的 Fraïssé 型理论，并证明具有有理距离的不可数可分度量空间的相关相容性及自同构群等结论。
Techniques of combinatorial set theory are applied to the following algebraic problem. Suppose G is an abelian group such that, for all countable subgroups C, the divisible part of the quotient G/C is countable. What can one conclude about the size of the divisible part of G/K when the cardinality of the subgroup K is a given uncountable cardinal?
Dense-set dependence in the Katětov order for uncountable coordinate ideals
该研究在Katětov序框架下，针对不可数坐标理想Conv(A,D)，在ZFC及CH公理下探讨其等价性与归约关系，证明CH下可数非小坐标界的严格性及非归约性
We indicate a way of distinguishing between structures, for which, we call two structures distinguishable. Roughly, being distinguishable means that they differ in the number of realizations each gives for some formula. Being non-distinguishable turns out to be an interesting equivalence relation that is weaker than isomorphism and stronger than elementary equivalence. We show that this equivalence relation is Borel in a Polish space that codes countable structures. It then follows, without assuming the Continuum Hypothesis, that for any first order theory in a countable language, if it has an…
Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever $\mathbf{A}$ is…
In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality $κ$ it is proved that this quasi-order has no maximal element for $κ$ at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. $2^κ$. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal …
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
Countable Borel treeable equivalence relations are classifiable by $\ell_1$
该研究通过自由巴拿赫空间归约证明可树化可数Borel等价关系可由阿贝尔波兰群分类，否定了Hjorth的相关猜想，同时证明可由局部紧阿贝尔波兰群可数积分类的CBER是超有限的，用到Hjorth波兰群作用分析方法。
Extremely amenable automorphism groups of countable structures
研究极可 amenable 群的数量及其作为可数结构自动同构群的构造，证明存在连续统多个非同构的极可 amenable 群。
A Galois correspondence for automorphism groups of structures with the Lascar Property
引入Lascar性质，研究可数饱和结构自同构群之间的拓扑同构，建立有限生成代数闭子集与其点稳定子群之间的伽罗瓦对应，并刻画ACF_0和无限维向量空间的自同构群。
A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.
In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in $ω^ω$. (III) There is an $L_{ω_1,ω}$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $κ$ of of cofinality $ω$ (cf. theorem 4.2). The following are equivalent: (I$^*$) There is a model of the Scott sentence of $M$ of size…
Cohen-like first order structures
本研究用力迫替代归纳法构造不可数泛型结构，证明其多为刚性，并给出自同构群为整数群的稠密实数集例子。
Forcing in bounded arithmetic and set theory: analogies and differences
本文概述有界算术中的多种力迫方法，与集合论力迫对比，指出可数框架可统一解释现有方法，并探讨类比能否发现新偏序。
Let G be a countable group. We proof that there is a model companion for the approximate theory of a Hilbert space with a group G of automorphisms. We show that G is amenable if and only if the structure induced by countable copies of the regular representation of G is existentially closed.
Our first motivation was the question: can a countable structure have an automorphism group, which a free uncountable group? This is answered negatively in [Sh:744]. Lecturing in a conference in Rutgers, February 2001, I was asked whether I am really speaking on Polish groups. We can prove this using a more restrictive condition on the set of equations. Parallel theorems, hold for semi groups and for metric algebras, e.g. with non-isolated unit. Here we do the general case. For instance we show that there is no Polish group which as a group is free and uncountable.
Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for the automorphism group of the rational order and the automorphism group of the random graph [Truss]. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.
On automorphisms of $\mathcal P(λ)/[λ]^{<λ}$
本文研究商布尔代数 $\mathcal P(λ)/[λ]^{<λ}$ 的自同构平凡性，分别在 MA、可测基数及不可达基数的力迫情形下给出一致性与反例结果。
In this paper we analyse some questions concerning trees on $κ$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in \cite[Question 5.2]{FKK16} about the diagram for regularity properties.
Simplicity of the automorphism group of fields with operators
本文改编 Lascar 的证明，证明了涵盖强极小理论和带算子域理论的一类不可数模型中，固定所有非泛型元素的自同构群具有单纯性。
A homogeneous family of subsets over a given set is one with a very ``rich'' automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of $2^\cont$ isomorphism types of homogeneous families over a countable set.
We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover various results of Kechris-Pestov-Todorčević, Moore, Ngyuen Van Thé, in the context of automorphism groups of not necessarily countable structures, as well as Zucker.
We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $Ω$, there are uncountably many maximal subgroups of $\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of $Ω$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning $X$. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal…
Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $ω_1$. We show it is consistent that $\mathcal{C}$ has minimal elements. This answers an old question due to Galvin.
A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a "geometric technique" used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman's theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment $\mathrm{KP…
We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.
Bowtie-free graphs and generic automorphisms
本文证明了可数泛型omega范畴无蝶形图允许泛型自同构，并指出该图不可有限齐化。
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
Let $G$ be a topological group and let $μ$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $μ$-equivalence classes of $μ$-measurable functions defined on [0,1] with values in $G$, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group $G$, if $L_0(G)$ has ample generics, then $G$ has ample generics, thus the converse to a result of Kaïchouh and Le Maître. We further study topological similarity classes and conjugacy classes for many groups ${\rm{Aut}}(M)$ and $L_0({\rm{Aut}}(M))$, w…
We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric s…
This paper shows that, even at the most basic level, the parallel, countable branching and uncountable branching recurrences of Computability Logic (see http://www.cis.upenn.edu/~giorgi/cl.html) validate different principles.
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