Olga Kharlampovich, Christopher Natoli
We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).
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We show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.
We analyze the forcing notion $\mathcal P$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_θ$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in $ω_1$.
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…
A result of Kaufmann shows that if $L_α$ is countable, admissible and satisfies $Π_n\textsf{-Collection}$, then $\langle L_α, \in \rangle$ has a proper $Σ_{n+1}$-elementary end extension. This paper investigates to what extent the theory that holds in $\langle L_α, \in \rangle$ can be transferred to the partially-elementary end extensions guaranteed by Kaufmann's result. We show that there are $L_α$ satisfying full separation, powerset and $Π_n\textsf{-Collection}$ that have no proper $Σ_{n+1}$-elementary end extension satisfying either $Π_{n}\textsf{-Collection}$ or $Π_{n+3}\textsf{-Foundatio…
We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact satura…
We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not $\forall$-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fraïssé classes in which embeddings must preserve $\forall$-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.
We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic tree…
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…
Partially-elementary end extensions of countable models of set theory
研究集合论可数模型的部分初等端扩展，证明在特定条件下存在满足递归理论的真Σ_n-初等端扩展，并由此推出M+Π_n-收集+Π_{n+1}-基础可证Σ_{n+1}-分离。
We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim-Skolem-Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber's quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show …
We introduce a simplified framework for ord-transitive models and Shelah's non elementary proper (nep) theory. We also introduce a new construction for the countable support nep iteration.
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.
It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable π-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.
We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(chi), in). This leads to forcing notions which are ``reasonably'' definable. We present two specific properties materializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness). For this we consider candidates (countable models to which the definition applies), and the older Souslin proper. A major theme here is ``preservation by iteration'', but we also show a dichotomy: if suc…
We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such $G$ include automorphism groups of countable $ω$-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of $G…
We investigate subsystems $COM_{fcn}$, $COMI_{fcn}$ and $PRA_{fcn}$ of the elementary theory of functions $ETF$, the base theory for countable strict reverse mathematics. We show that inductions on any variable for unary, binary and ternary functions are pairwise equivalent over $COM_{fcn}$. We prove that weakened primitive recursion axiom $WPRA$ is equivalent to primitive recursion axiom $PRA$ over $COMI_{fcn}$. We show that permutation axiom and minimization axioms $MIN^1$, $MIN^2$, $MIN^3$ are pairwise equivalent over $PRA_{fcn}$. Thus, we present several equivalent axiomatizations of $ETF$…
Finite-index extensions of essentially free countable Borel equivalence relations
本研究针对Kechris提出的可数Borel等价关系本质自由性的有限指数扩张保持性问题，证明E本质自由时其对应的E^fin也本质自由，从而给出该问题的肯定答案。
Models of an Abstract Elementary Class as a Generalized Polish Space
本文利用 Baldwin-Boney 关系表示定理，将一阶逻辑中模型空间为波兰空间的结果推广到抽象初等类，证明其模型在特定条件下构成广义波兰空间。
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.
We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, we give an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.
We show that any countable model of a model complete theory has an elementary extension with a "pseudofinite-like" quasidimension that detects dividing.
In this note we give a new proof of the fact that an elementary subgroup (in the sense of first-order theory) of a non abelian free group $\mathbb{F}$ must be a free factor. The proof is based on definability of orbits of elements of under automorphisms of $\mathbb{F}$ fixing a large enough subset of $\mathbb{F}$.
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory $T$ is not $ω$-stable, then the elementary diagram of some countable model of $T$ has a Borel complete reduct.
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…
We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups elementarily embedded in finitely generated free groups are free factors.
We prove that two co-Hopfian finitely generated virtually free groups are elementarily equivalent if and only if they are isomorphic. We also prove that co-Hopfian finitely generated virtually free groups are homogeneous in the sense of model theory.
Groups elementary equivalent to finitely generated free metabelian
本文描述了初等等价于具有n个生成元的自由亚幂零群的群，并探讨了亚幂零群中自然出现的指数运算。
This paper is the 10th in a sequence on the structure of sets of solutions to systems of equations over groups, projections of such sets (Diophantine sets), and the structure of definable sets over few classes of groups. In the 10th paper we answer affirmatively a question of R. L. Vaught on the elementary equivalence of free products of pairs of elementarily equivalent groups, and obtain a generalization of Tarski's problem on the elementary equivalence of non-abelian free groups. Finally, we prove that free products of stable groups is stable, generalizing a previous result on the stability …
Uniform first order interpretation of the second order theory of countable groups of homeomorphisms
该研究证明了流形同胚群的一阶理论可统一解释其二阶理论，并揭示了同胚群的可定义子集及相关计算理论结果。
Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order
本文证明了无 torsion 的 Abel 群是忠实 Borel 完整的，并展示了纯可嵌入性是一个完全的解析准序。
We prove that there is no countable universal $B_n$-free graph for all $n$ and that there is no countable universal graph in the class of graphs omitting all cycles of length at most $2k$ for $k\ge 2$.
We show that for any infinite countable group $G$ and for any free Borel action $G \curvearrowright X$ there exists an equivariant class-bijective Borel map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G \curvearrowright 2^G$. This implies that any Borel structurability which holds for the equivalence relation generated by $G \curvearrowright \mathrm{Free}(2^G)$ must hold a fortiori for all equivalence relations coming from free Borel actions of $G$. A related consequence is that the Borel chromatic number of $\mathrm{Free}(2^G)$ is the maximum among Borel chromatic numbers…
Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an aleph_1-free R-module G of rank aleph_1 with endomorphism algebra End_RG=A . Clearly the result does not hold for fields. Recall that an R-module is aleph_1-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra A in use. For instance, if we choose A=R, then clearly G is an indecomposable `almost free' module. The existence of such modules …
A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countabl…
We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is Borel complete. In the process, we also rediscover the main result of [7] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.
We show that in algebraically locally finite countable homogeneous structures with a free stationary independence relation the small index property implies the strong small index property. We use this and the main result of [15] to deduce that countable free homogeneous structures in a locally finite relational language have the strong small index property. We also exhibit new continuum sized classes of $\aleph_0$-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.
We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria for such classes to be abstract elementary classes, for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translate into a machinery to build concrete examples of $\mathrm{AECs}$ satisfying given model-theor…
Equivalents of NOTOP
本文在可数超稳定理论中给出 NOTOP 的多种等价刻画，证明其等价于 V-DI，并验证了 Shelah 关于 NOTOP 蕴含 PMOP 的断言。
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。
We modify the notion of a Fraïssé class and show that various interesting classes of groups, notably the class of nonabelian limit groups and the class of finitely generated elementary free groups, admit Fraïssé limits. Furthermore, we rediscover Lyndon's $\Z[t]$-exponential completions of countable torsion-free CSA groups, as Fraïssé limits with respect to extensions of centralizers. Dedicated to the memory of Charles Sims.
We provide a co-free construction which adds elementary structure to a primary doctrine. We show that the construction preserves comprehensions and all the logical operations which are in the starting doctrine, in the sense that it maps a first order many-sorted theory into a the same theory formulated with equality. As a corollary it forces an implicational doctrine to have an extentional entailment.
Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $μ$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $Γ$, there is a Borel sequence $(\cdot_r)_{r \in \mathbb{R}}$ of free actions of $Γ$ on $X$, generating subequivalence relations $E_r$ of $E$ with respect to which $μ$ is ergodic, with the further property that $(E_r)_{r \in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $μ$-reducibility. In particular, we show that if…
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…
Equivalence of Random Generics Is Not Essentially Free
该研究针对ZFC模型上随机泛型的等价关系，结合Popa超刚性等工具证明其正测度限制不可弱归约到本质自由等价关系。
Dichotomy results for classes of countable graphs
本文研究了可数图类中排除特定有限图作为诱导子图的结构分化，揭示了当排除图与P4相关时，该类别的计算行为和结构复杂性存在显著差异。
We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide several new examples of weak Fraïssé classes of finite graphs. Finally, we show that the class of all countable $C_4$-free graphs does not contain a generic structure, strengthening a result of A. Hajnal and J. Pach.
We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning …
We define the standard Borel space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $τ$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.
We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^ω/ \mathcal{U} \simeq \mathbf{N}^ω/ \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $ω,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.
Weight, net weight, and elementary submodels
本文通过初等子模型方法，在ZFC下构造了$C(ω_1)$中净权重不可数的Hausdorff空间，并证明添加Cohen实数后正则空间净权重可数，同时解决了文献[6]中的两个主要问题。
We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation induced by the natural action of $\mathrm{GL}_2(\mathbb{Q})$ on $\mathrm{Ar}(\mathbb{Q}^2)$ is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau. In particular, we show that its potential class is not $\boldsymbolΠ^0_…
Elementary properties of free lattices III: Undecidability of the full theory
针对此前遗留的无限自由格完备理论可判定性开放问题，本研究证明对任意基数$κ\geq3$，自由格$\mathbf{F}_κ$的一阶理论不可判定，完成了该问题的解答。
We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB, the class of torsion-free abelian groups. We also prove that for any countable ring $R$, both the class of left $R$-modules endowed with an endomorphism and the class of left $R$-modules with four named submodules are Borel complete.
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。
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