Essential $\aleph_0$-categoricity; i.e., $\aleph_0$-categoricity in some full countable language, is shown to be a robust notion for strongly minimal compact complex manifolds. Characterisations of triviality and essential $\aleph_0$-categoricity are given in terms of complex-analytic automorphisms, in the simply connected case, and correspondences in general. As a consequence it is pointed out that an example of McMullen yields a strongly minimal compact Kähler manifold with trivial geometry but which is not $\aleph_0$-categorical, giving a counterexample to a conjecture of the second author and Tom Scanlon.
We generalise the correspondence between $\aleph 0$-categorical theories and their automorphism groups to arbitrary complete theories in classical logic, and to some theories (including, in particular, all $\aleph 0$-categorical ones) in continuous logic.
We provide a description of the structure of $\aleph_0$-categorical trees and cycle-free partial orders. First the maximal branches of $\aleph_0$-categorical tree are examined, followed by the configuration of the ramification orders, which are then combined to provided necessary and sufficient conditions for a tree to be $\aleph_0$-categorical in terms of these two things. The classification of the $\aleph_0$-categorical cycle-free partial orders is found as a corollary.
These notes find a canonical representation of the $\aleph_0$-categorical linear orders based on Joseph Rosenstein's description. A unique minimal representation, called the normal form, is obtained.
We extend the Ahlbrandt--Ziegler analysis of interpretability in aleph_0-categorical structures by showing that existential interpretation is controlled by the monoid of self--embeddings and positive existential interpretation of structures without constant endomorphisms is controlled by the monoid of endomorphisms in the same way as general interpretability is controlled by the automorphism group.
Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain {a: M models P(a)}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then there is an isomorphism i:M-> N which is the identity on P(M). T is relatively categorical if it is relatively lambda-categorical for every lambda. The question arises whether the relative lambda-categoricity of T for s…
We embed a countably categorical group G into a locally compact group c(G) with a non-trivial topology and study how topological properties of c(G) are connected with the structure of definable subgroups of G.
We classify the computability-theoretic complexity of two index sets of classes of first-order theories: We show that the property of being an $\aleph_0$-categorical theory is $Π^0_3$-complete; and the property of being an Ehrenfeucht theory $Π^1_1$-complete. We also show that the property of having continuum many models is $Σ^1_1$-hard. Finally, as a corollary, we note that the properties of having only decidable models, and of having only computable models, are both $Π^1_1$-complete.
A countable semigroup is $\aleph_0$-categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the $\aleph_0$-categoricity of semigroups. Our main results are a complete classification of $\aleph_0$-categorical orthodox completely 0-simple semigroups, and descriptions of the $\aleph_0$-categorical members of certain classes of strong semilattices of semigroups.
We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the know…
Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …
A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, in which $R$ can be embedded. Let $sib(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number…
We study approximate $\aleph_0$-categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for which the corresponding theory $Q_P$ of beautiful pairs is not approximately $\aleph_0$-categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory $Q$ of randomized infinite vector spaces over a finite field is such an example. On th…
This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) $M_{\mathcal{T}}$ has the \emph{non independence property} (NIP); (2) every Banach space that is $\aleph_0$-categorical up to small perturbations embeds $c_0$ or $\ell_p$ ($1\leqslant p<\infty$) almost isometrically; consequ…
In this paper we initiate the study of $\aleph_0$-categorical semigroups, where a countable semigroup $S$ is $\aleph_0$-categorical if, for any natural number $n$, the action of its group of automorphisms Aut $S$ on $S^n$ has only finitely many orbits. We show that $\aleph_0$-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between $\aleph_0$-categoricity and a number of semigroup and monoid constructions, namely direct sums, 0-direct unions, semidirect products and $\mathcal{P}$-semigroups. As a corollary, w…
We study automorphism groups of randomizations of separable structures, with focus on the $\aleph_0$-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the $\aleph_0$-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and cl…
We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0. Another central result here is, in this context: the number of models of a countable first order T of cardinality aleph_alpha is either >=|alpha| for every aleph_αor it has a small upper bound (close to beth_2).
We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective $R$-modules is $λ$-categorical in $all$ $λ> |R|+\aleph_0$ if and only if $R \cong M_n(D)$ for $D$ a division ring and $n \geq 1$. 2. The class of flat $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R \cong M_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $n \geq 1$. 3. Assume $R$ is a commutative ring.…
We investigate the automorphism groups of $\aleph\_0$-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable $\aleph\_0$-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and c…
Diestel and Leader have characterised connected graphs that admit a normal spanning tree via two classes of forbidden minors. One class are Halin's $(\aleph_0,\aleph_1)$-graphs: bipartite graphs with bipartition $(\mathbb{N},B)$ such that $B$ is uncountable and every vertex of $B$ has infinite degree. Our main result is that under Martin's Axiom and the failure of the Continuum Hypothesis, the class of forbidden $(\aleph_0,\aleph_1)$-graphs in Diestel and Leader's result can be replaced by one single instance of such a graph. Under CH, however, the class of $(\aleph_0,\aleph_1)$-graphs contain…
We give necessary and sufficient conditions for a polynomially bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be $\aleph_α$-saturated. The conditions are in terms of the value group, residue field, and pseudo- Cauchy sequences of the natural valuation on the real closed field. This is achieved by an analysis of types, leading to the trichotomy. Our characterization provides a construction method for saturated models, using fields of generalized power series.
We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise…
We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is $\aleph_0$-categorical and describe definable and algebraic closures in that theory. We also obtain similar results for Boole-De Morgan algebras.
The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Ban…
A linear order $L$ is strongly surjective if $L$ can be mapped onto any of its suborders in an order preserving way. We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of Camerlo, Carroy and Marcone. In particular, $\diamondsuit^+$ implies the existence of a lexicographically ordered Suslin-tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under $2^{\aleph_0}<2^{\aleph_1}$ or in the Cohen and other canonical models (where $2^{\aleph_0}=2^{\aleph_1}$); fina…
We prove that $\ omega $-categorical dp-minimal groups are nilpotent-by-finite. We also show that in dp-minimal definably amenable groups, f-generic global types are strongly f-generic.
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；University of Torino（都灵大学）
A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner $k$-system (for $k \geq 2$) is a linear space such that each line has size exactly $k$. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary $τ$ with a single ternary relation $R$. We prove that for every integer $k$ there exist $2^{\aleph_0}$-many integer valued functions $μ$ such that each $μ$ determines a distinct strongly minimal Steiner $k$-system $\mathcal{G}_μ$, whose algebraic closure ge…
Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the set $U:=\{ y:E(t,y) \wedge y' \neq 0 \}$ where $t$ is an element with $t'=1$. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U$. Moreover, the induced structure on Cartesian powers of $U$ is given by special subvarieties. If $E$ has some special form then all fibres $U_s:=\{ y:E(s,y) \wedge y' \neq 0 \}$ (with $s…
