Narrow Results

A first-order theory is Noetherian with respect to the collection of formulae ${ℱ}$ if every definable set is a Boolean combination of instances of formulae in ${ℱ}$ and the topology whose subbasis of closed sets is the collection of instances of arbitrary formulae in ${ℱ}$ is Noetherian. We show the Noetherianity of the theory of proper pairs of algebraically closed fields in any characteristic with respect to the family of tame formulae as introduced in [A. Martin-Pizarro and M. Ziegler, Equational theories of fields, J. Symbolic Logic85 (2020) 828–851, https://arxiv.org/abs/1702.05735], thus answering a question which was left open there.

We initiate the study of a generalization of Kim-independence, Conant-independence, based on the notion of strong Kim-dividing of Kaplan, Ramsey and Shelah. A version of Conant-independence was originally introduced to prove that all ${\mathrm{\text{NSOP}}}_2$ theories are ${\mathrm{\text{NSOP}}}_1$. We introduce an axiom on stationary independence relations, essentially generalizing the “freedom” axiom in some of the free amalgamation theories of Conant, and show that this axiom provides the correct setting for carrying out arguments of Chernikov, Kaplan and Ramsey on ${\mathrm{\text{NSOP}}}_1$ theories relative to a stationary independence relation. Generalizing Conant’s results on free amalgamation to the limits of our knowledge of the ${\mathrm{\text{NSOP}}}_n$ hierarchy, we show using methods from Conant as well as our previous work that any theory where the equivalent conditions of this local variant of ${\mathrm{\text{NSOP}}}_1$ holds is either ${\mathrm{\text{NSOP}}}_1$ or ${\mathrm{\text{SOP}}}_3$ and is either simple or ${\mathrm{\text{TP}}}_2$. We observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric. This includes all of Conant’s examples, the small cycle-free random graphs of Shelah, and the (finite-language) $\omega$-categorical Hrushovski constructions of Evans and Wong. We then answer Conant’s question on the existence of non-modular free amalgamation theories. We show that the generic functional structures of Kruckman and Ramsey are examples of non-modular free amalgamation theories. We also show that any free amalgamation theory is ${\mathrm{\text{NSOP}}}_1$ or ${\mathrm{\text{SOP}}}_3$, while an ${\mathrm{\text{NSOP}}}_1$ free amalgamation theory is simple if and only if it is modular. Finally, we show that every theory where Conant-independence is symmetric is ${\mathrm{\text{NSOP}}}_4$. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the ${\mathrm{\text{NSOP}}}_n$ hierarchy beyond ${\mathrm{\text{NSOP}}}_1$. We explain the connection to some established open questions.

How can non-classical logic contribute to the analysis of complexity in computer science? In this paper, we give a step towards this question, taking a logical model-theoretic approach to the analysis of complexity in fuzzy constraint satisfaction. We study fuzzy positive-primitive sentences, and we present an algebraic characterization of classes axiomatized by this kind of sentences in terms of homomorphisms and direct products. The ultimate goal is to study the expressiveness and reasoning mechanisms of non-classical languages, with respect to constraint satisfaction problems and, in general, in modelling decision scenarios.

We associate with every first order structure a family of invariant, locally Noetherian topologies (one topology on each Mⁿ). The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper, we explore stability results in this new context. We assume that is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:.

Theorem 0.1. Suppose that is not only tame, but -tame. If and is Galois stable in μ, then , where is a relative of κ(T) from first order logic.

is the Hanf number of the class . It is known that .

The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting (improving [22, Claim 4.15] and a result from [7]) and the following initial step towards a stability spectrum theorem for tame classes:.

Theorem 0.2. If is Galois-stable in some , then is stable in every κ with κ^μ=κ. For example, under GCH we have that Galois-stable in μ implies that is Galois-stable in μ⁺ⁿ for all n < ω.

Let T₁ and T₂ be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T₁ ∪ T₂ has a strongly minimal completion.

We prove that from categoricity in λ⁺ we can get categoricity in all cardinals ≥ λ⁺ in a χ-tame abstract elementary classe which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided and λ ≥ χ.

For the missing case when , we prove that is totally categorical provided that is categorical in and .

Summary. From known examples of theories T obtained by Hrushovski-constructions and of infinite Morley rank, properties are extracted, that allow the collapse to a finite rank substructure. The results are used to give a more model-theoretic proof of the existence of the new uncountably categorical groups in [3].

We provide a new criterion for embedding 𝔼₀, and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on the Borel cardinality of Lascar strong types equality, and Newelski's results about pseudo F_σ groups.

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the $(p,q)$-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of $M$-invariant types to that of $M$-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding $A^{♯}=p^{\infty }A({𝒰})$ where ${𝒰}$ is a saturated separably closed field.

We classify countable metrically homogeneous graphs of diameter 3.

Let $G$ be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in $m$ commuting derivations equipped with a $G$-action by differential field automorphisms. In the language of $G$-differential rings (i.e. the language of rings with added symbols for derivations and automorphisms), we prove that this class has a model-companion — denoted $G-{\mathrm{\text{DCF}}}_{0,m}$. We then deploy the model-theoretic tools developed in the first author’s paper [D. M. Hoffmann, Model theoretic dynamics in a Galois fashion, Ann. Pure Appl. Logic 170(7) (2019) 755–804] to show that any model of $G-{\mathrm{\text{DCF}}}_{0,m}$ is supersimple (but unstable when $G$ is nontrivial), a PAC-differential field (and hence differentially large in the sense of the second author and Tressl [Differentially large fields, preprint (2020), arXiv:2005.00888, available at https://arxiv.org/abs/2005.00888]), and admits elimination of imaginaries after adding a tuple of parameters. We also address model-completeness and supersimplicity of theories of bounded PAC-differential fields (extending the results of Chatzidakis and Pillay [Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998) 71–92] on bounded PAC-fields).

When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map $\text{res}:k((t))\to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for t. Driven by this observation, we study the theory ${\text{VF}}_{\text{res},\iota }$ of valued fields equipped with a linear form $\text{res}:K\to k$ which restricts to the residue map on the valuation ring. We prove that ${\text{VF}}_{\text{res},\iota }$ does not admit a model companion. In addition, we show that $(k((t)),\text{res})$ is undecidable whenever k is an infinite field. As a consequence, we get that $(ℂ((t)),{\text{Res}}_0)$ is undecidable, where ${\text{Res}}_0:f\mapsto {\text{Res}}_0(f)$ maps f to its complex residue at 0.

We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals.

We give a description of solutions of polynomial-exponential equations where the variables vary over rational numbers. Using this, we also prove a finiteness result.

We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in the group is a union of a family of quantifier free definable sets, where the parameter of the family runs over a set definable (with quantifiers) in a sort which carries the structure of an ordered set with some additional unary predicates.

As a corollary, we find that all definable functions in ordered abelian groups are piecewise linear on finitely many definable pieces.

We survey some results on Banach spaces and related structures, viewed as first-order structures in the model-theoretic setting. We study some geometric properties defined by the neocompact formulas: back-and-forth, density, quantifier elimination, winning strategy. Various kinds of saturation (compactness) are considered. Included also are applications to definable mappings, fixed point theorem and the Banach-Mazur problem on characterizing Hilbert spaces.

In this paper, we turn our attention to model theory of higher-order fuzzy logic (fuzzy type theory). This theory generalizes model theory of predicate logic but has some interesting specificities. We will introduce few basic concepts related to homomorphism, isomorphism, submodel, etc. and show some properties of them.