We investigate the primitive recursive content of linear orders. We prove that the punctual degrees of rigid linear orders, the order of the integers $\mathbb {Z}$, and the order of the rationals $\mathbb {Q}$ embed the diamond (preserving supremum and infimum). In the cases of rigid orders and the order $\mathbb {Z}$, we further extend the result to embed the atomless Boolean algebra; we leave the case of $\mathbb {Q}$ as an open problem. We also show that our results for the rigid orders, in fact, work for orders having a computable infinite invariant rigid sub-order.

We prove the existence of a model companion of the two-sorted theory of c-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field sort. Using a new criterion which does not rely on a stationary independence relation, we prove that if the field is NSOP$_1$, then the model companion is NSOP$_4$. We also prove that if the field is algebraically closed, then the model companion is c-NIP.

We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra $\mathcal {M}$ is never model complete if its direct integral decomposition contains $\mathrm {II}_1$ factors $\mathcal {N}$ such that $M_2(\mathcal {N})$ embeds into an ultrapower of $\mathcal {N}$. The proof in the case of $\mathrm {II}_1$ factors uses an explicit construction based on random matrices and quantum expanders.

We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals of tracial factors implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś’s theorem and countable saturation, to this more general setting.

In the classification of complete first-order theories, many dividing lines have been defined in order to understand the complexity and the behavior of some classes of theories. In this paper, using the concept of patterns of consistency and inconsistency, we describe a general framework to study dividing lines and introduce a notion of maximal complexity by requesting the presence of all the exhibitable patterns of definable sets. Weakening this notion, we define new properties (Positive Maximality and the $\mathrm {PM}^{(k)}$ hierarchy) and prove some results about them. In particular, we show that $\mathrm {PM}^{(k+1)}$ theories are not k-dependent. Moreover, we provide an example of a $\mathrm {PM}$ but $\mathrm {NSOP}_4$ theory (showing that $\mathrm {SOP}$ and the $\mathrm {SOP}_n$ hierarchy, for $n \geq 4$, cannot be described by positive patterns) and, for each $1<k<\omega $, an example of a $\mathrm {PM}^{(k)}$ but $\mathrm {NPM}^{(k+1)}$ theory (showing that the newly defined hierarchy does not collapse).

We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of ${\mathbb N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and $\mathrm {C}^{*}$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.

We introduce the notion of echeloned spaces – an order-theoretic abstraction of metric spaces. The first step is to characterize metrizable echeloned spaces. It turns out that morphisms between metrizable echeloned spaces are uniformly continuous or have a uniformly discrete image. In particular, every automorphism of a metrizable echeloned space is uniformly continuous, and for every metric space with midpoints, the automorphisms of the induced echeloned space are precisely the dilations.

Next, we focus on finite echeloned spaces. They form a Fraïssé class, and we describe its Fraïssé-limit both as the echeloned space induced by a certain homogeneous metric space and as the result of a random construction. Building on this, we show that the class of finite ordered echeloned spaces is Ramsey. The proof of this result combines a combinatorial argument by Nešetřil and Hubička with a topological-dynamical point of view due to Kechris, Pestov and Todorčević. Finally, using the method of Katětov functors due to Kubiś and Mašulović, we prove that the full symmetric group on a countable set topologically embeds into the automorphism group of the countable universal homogeneous echeloned space.

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in q. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed).

The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma $ which is locally infinitely contracting and fails to be onto. Namely, we prove the existence of a model complete theory $\widetilde {\mathrm {VFE}}$ amalgamating the theories $\mathrm {SCFE}$ and $\widetilde {\mathrm {VFA}}$ introduced in [5] and [11], respectively. In characteristic zero, we also prove that $\widetilde {\mathrm {VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the value group.

Using motivic integration theory and the notion of riso-triviality, we introduce two new objects in the framework of definable nonarchimedean geometry: a convenient partial preorder $\preccurlyeq$ on the set of constructible motivic functions, and an invariant $V_0$, nonarchimedean substitute for the number of connected components. We then give several applications based on $\preccurlyeq$ and $V_0$: we obtain the existence of nonarchimedean substitutes of real measure geometric invariants $V_i$, called the Vitushkin variations, and we establish the nonarchimedean counterpart of a real inequality involving $\preccurlyeq$, the metric entropy and our invariants $V_i$. We also prove the nonarchimedean Cauchy–Crofton formula for definable sets of dimension $d$, relating $V_0$ (and $V_d$) and the motivic measure in dimension $d$.

A first-order expansion of $(\mathbb {R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, p-adic fields, ordered abelian groups with only finitely many convex subgroups (in particular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb {Z},+,S)$, where S is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb {Q},+,<)$ and o-minimal expansions ${\mathscr R}$ of $(\mathbb {R},+,<)$ such that $({\mathscr R},\mathbb {Q})$ is a “dense pair.”.

Pre-H-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-H-fields that are differential-Hensel–Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax–Kochen/Ershov theorem for such structures: the theory of a differential-Hensel–Liouville closed pre-H-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel–Liouville closed pre-H-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-H-field sort, from which we deduce that the ordered differential residue field is purely stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel–Liouville closed pre-H-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-H-fields with gap $0$, and is complete, distal, and locally o-minimal.

Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment $\mathcal {V}$. In this context, we construct atomic formulas and define the regular fragment of our enriched logic by taking conjunctions and existential quantification of those. We then characterize $\mathcal {V}$-categories of models of regular theories as enriched injectivity classes in the $\mathcal {V}$-category of structures. These notions rely on the choice of an orthogonal factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ which will be used, in particular, to interpret relation symbols and existential quantification.

We prove that any compact R-analytic group is linear when R is a pro-p domain of characteristic zero.

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, $C^*$-algebras and the measure algebra of an infinite compact group.

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in \mathbb {N}$ and any countable model of $\mathrm {B}\Sigma _{n+2}$, we construct a proper $\Sigma _{n+2}$-elementary end extension satisfying $\mathrm {B}\Sigma _{n+1}$, which answers a question by Clote positively. We also give a characterization of the countable models of $\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of $\mathrm {B}\Sigma _{n+2}$. Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.

There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include $\{\mathord {\downarrow }, \mathord {@}\}$ with or without nominals (Areces, Blackburn, Marx), $\mathord {@}$ with or without nominals (ten Cate), and $\mathord {\downarrow }$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation theorem for all standard fragments of hybrid logic, in particular for the case with $\mathord {\downarrow }$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense.

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we extend this result to $\kappa $-prime models, for $\kappa $ an uncountable cardinal or $\aleph _\varepsilon $.

We study H-structures associated with $SU$-rank 1 measurable structures. We prove that the $SU$-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

Let $\operatorname {TFAb}_r$ be the class of torsion-free abelian groups of rank r, and let $\operatorname {FD}_r$ be the class of fields of characteristic $0$ and transcendence degree r. We compare these classes using various notions. Considering the Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the $\operatorname {TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname {FD}_r$. Thomas and Velickovic showed that for sufficiently large r, the classes $\operatorname {FD}_r$ are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of $\operatorname {TFAb}_r$ in $\operatorname {FD}_r$, and of $\operatorname {FD}_r$ in $\operatorname {FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname {TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname {TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma _3$.