We investigate this class of groups originally called ulf (universal locally finite groups) of cardinality $λ$. We prove that for every locally finite group $G$ there is a canonical existentially closed extention of the same cardinality, unique up to isomorphism and increasing with $G$. Also we get, e.g. existence of complete members (i.e. with no non-inner automorphisms) in many cardinals (provably in ZFC). We also get a parallel to stability theory in the sense of investigating definable types.
This paper deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author. Here we study this class in Robinson's logic, meaning the category of existentially closed models with embeddings following Haykazyan and Kirby, and prove that in this context this class is NSOP$_1$ and TP$_2$.
We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. We extend some notions from classification theory to positive logic and position the category of existentially closed exponential fields in the stability hierarchy as NSOP$_1$ but TP$_2$.
In the first part of the paper we survey several results from Popa's deformation/rigidity theory on the classification of tensor product decompositions of large natural classes of II$_1$ factors. Using a mélange of techniques from deformation/rigidity theory, model theory, and the recent works \cite{CIOS21,CDI22} we highlight an uncountable family of existentially closed II$_1$ factors $M$ which do not admit tensor product decompositions $M= P\bar \otimes Q$ into diffuse factors where $Q$ is full. In the last section we discuss several open problems regarding the structural theory of existenti…
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.
Let $G$ be a finitely presented group, and let $H$ be a subgroup of $G$. We prove that if $H$ is acylindrically hyperbolic and existentially closed in $G$, then $G$ is acylindrically hyperbolic. As a corollary, any finitely presented group which is existentially equivalent to the mapping class group of a surface of finite type, to $\mathrm{Out}(F_n)$ or $\mathrm{Aut}(F_n)$ for $n\geq 2$ or to the Higman group, is acylindrically hyperbolic.
We study several model-theoretic aspects of W$^*$-probability spaces, that is, $σ$-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W$^*$-spaces and prove several structural results about such spaces, including that they are type III$_1$ factors that tensorially absorb the Araki-Woods factor $R_\infty$. We also study the existentially closed objects in the restricted class of W$^*$-probability spaces with Kirchberg's QWEP property, proving that $R_\infty$ itself is such an existentially closed space in this class. Our results about exis…
The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$ does not interpret Robinson's theory $R$. To this end, we borrow tools from model theory--specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of $\exists\forall$ theories interpretable in existential theories in the process.
We show that there is $n\in \mathbf N$, a finite system $Σ(\vec x,\vec y)$ of equations and inequations having a solution in some group, where $\vec x$ has length $n$, and $ε>0$ such that: for any group $G$ and any $\vec a\in G^n$, if the system $Σ(\vec a,\vec y)$ has a solution in $G$, then there is no $(\vec a,ε)$-Fø lner set in $G$. The proof uses ideas from model-theoretic forcing together with the observation that no amenable group can be existentially closed. Along the way, we also observe that no existentially closed group can be exact, have the Haagerup property, or have property (T). …
We investigate existentially closed models (of a quite arbitrary theory) equipped which an action of a fixed group G. We embed these structures in a monster model D of some well-rounded theory and describe them as PAC substructures of D. Assuming that the class of these existentially closed models is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas - similar to ACFA. Moreover, it codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imagina…
We study the theory $T_{m,n}$ of existentially closed incidence structures omitting the complete incidence structure $K_{m,n}$, which can also be viewed as existentially closed $K_{m,n}$-free bipartite graphs. In the case $m = n = 2$, this is the theory of existentially closed projective planes. We give an $\forall\exists$-axiomatization of $T_{m,n}$, show that $T_{m,n}$ does not have a countable saturated model when $m,n\geq 2$, and show that the existence of a prime model for $T_{2,2}$ is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model the…
In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA,the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to comm…
We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and $p$CF.
