arXiv subjects
Publications and source records attributed to Christian d'Elbée.
We prove that NIP nilrings of bounded nilexponent are nilpotent. We also deduce that NIP $ω$-categorical $n$-Engel Lie algebras of characteristic $p>n$ are nilpotent, yielding a case of the Lie algebra analogous of Wilson conjecture for $ω$-categorical groups.
In earlier work, the authors gave a construction and description of an amalgamated free product of filtered Lie algebras within a fixed nilpotency class, based on an intricate induction rooted in the work of Maier and of Higman. In this paper, the authors give a new description of this amalgam, adopting the viewpoint and methods from the work of A. I. Shirshov on amalgamation of Lie algebras, substantially simplifying their previous approach. This new description yields both group-theoretic and descriptive set-theoretic applications. A $c$-nilpotent group is called UL-equivalent if its lower and upper central series coincide. We prove that every finite $c$-nilpotent group of prime exponent $p$ with $p>c$ embeds in a finite UL-equivalent $c$-nilpotent group of exponent $p$. This recovers a result due to Ivanov and Majcher that the Polish space of enumerated $c$-nilpotent groups of exponent $p>c$ has a comeager orbit. Our result also has the following consequences. First, the class of finite $c$-nilpotent groups of exponent $p>c$ has the \textit{cofinal} amalgamation property (answering a question of Ivanov and Majcher, who showed that it has the \textit{weak} amalgamation property). Second, the reduct of the Fraïssé limit of $c$-Lazard groups of exponent $p>c$ to the group language is generic in the space of enumerated groups. Finally, we prove analogues of these results also for torsion-free $c$-nilpotent groups.
We are interested in the following conjecture of Wilson from 1981: every locally nilpotent countably categorical group is nilpotent. Following our previous work on the Lie algebra analogue of the conjecture, we use Lie methods to deduce the first nontrivial cases of the Wilson conjecture for n-Engel groups: countably categorical 3-Engel groups and 4-Engel 5-groups are nilpotent, from which we also conclude that countably categorical 4-Engel groups of odd exponent are nilpotent. This is implemented via an exceptional case of the Lazard correspondence, checked using computer algebra systems. We also study the transfer of nilpotency results between the three categories: groups, Lie algebras, associative algebras. Among other things, we prove that the Wilson conjecture implies the analogous nilpotency statement for associative algebras (modulo the commutative case).
This article is written in celebration of the 8th Kazakh-French Logical Colloquium. We expand on an unpublished research note of the second author. We record some results concerning local Keisler measures with respect to a formula which is stable in a model. We prove that in this context, every local Keisler measure on the associated local type space is a weighted sum of (at most countably many) types. Using this observation, we give an elementary proof of the commutativity of the Morley product in this context. We then give a functional analytic proof that the double limit property lifts to the appropriate evaluation map on pairs of local measures. We end with some comments on the NOP and local measures in the (properly) stable context.
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.
This course introduces the fruitful links between model theory and a combinatoric of sets given by independence relations. An independence relation on a set is a ternary relation between subsets. Chapter 1 should be considered as an introductory chapter. It does not mention first-order theories or formulas. It introduces independence relations in a naive set theory framework. Its goal is to get the reader familiar with basic axioms of independence relations (which do not need an ambient theory to be stated) as well as introduce closure operators and pregeometries. Chapter 2 introduces the model-theoretic context. The two main examples (algebraically closed fields and the random graph) are described as well as independence relations in those examples. Chapter 3 gives the axioms of independence relations in a model-theoretic context. It introduces the general toolbox of the model-theorists (indiscernible sequences, Ramsey/Erdos-Rado and compactness) and the independence relations of heirs/coheirs with two main applications: Adler's theorem of symmetry (how symmetry emerges from a weaker set of axioms, which is rooted in the work of Kim and Pillay) and a criterion for NSOP4 using stationary independence relations in the style of Conant. Independence relations satisfying Adler's theorem of symmetry are here called 'Adler independence relations' or AIR. Chapter 4 treats forking and dividing. It is proved that dividing independence is always stronger than any AIR (even though it is not an AIR in general) a connection between the independence theorem and forking independence, which holds in all generality and is based on Kim-Pillay's approach. Then, simplicity is defined and the interesting direction of the Kim-Pillay theorem (namely that the existence of an Adler independence relation satisfying the independence theorem yields simplicity) is deduced from earlier results.
We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative map, which we call ACFH. Among others, we prove that this theory is NSOP$_1$ and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.
We continue our study of the Wilson conjecture for $ω$-categorical Lie algebras and prove that $ω$-categorical $4$-Engel Lie algebras of characteristic $3$ are nilpotent. We develop a set of tools to adapt in the definable context some classical methods for studying Engel Lie algebras (Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others). We solve the case at hand by starting a systematic study of Lie algebras for which there is a $k$ such that the principal ideal generated by any element is nilpotent of class $<k$ (which we call $k$-strong Lie algebras). We use computer algebra to check basic cases of a conjectural arithmetical property of those, namely that $x^{k-1}y^{k-1} = (-1)^{k-1}y^{k-1}x^{k-1}$ is an identity for Lie elements of the enveloping algebra. The solution is given by reducing the problem to $k$-strong Lie algebras generated by particularly well behaved sandwiches in the sense of Kostrikin.
We prove a version of the Wilson conjecture for $ω$-categorical $3$-Engel Lie algebras over a field of characteristic $5$: every $ω$-categorical Lie algebra over $\mathbb{F}_5$ which satisfies the identity $[x,y^3] = 0$ is nilpotent. We also include an extended introduction to Wilson's conjecture: \textit{every $ω$-categorical locally nilpotent $p$-group is nilpotent}, and present variants of this conjecture and connections to local/global nilpotency problems (Burnside, Kurosh-Levitzki, Engel groups). No particular knowledge of model theory is assumed except basic notions of formulas and definable sets.
We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent $p$ studied by Baudisch is 2-dependent and NSOP$_{1}$. We prove that the class of $c$-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for $2 < c$, the generic $c$-nilpotent Lie algebra over $\mathbb{F}_{p}$ is strictly NSOP$_{4}$ and $c$-dependent. Via the Lazard correspondence, we obtain the same result for $c$-nilpotent groups of exponent $p$, for an odd prime $p > c$.
We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from the following construction: there is a dp-minimal valuation overring O extending R, a proper ideal I in O, and a finite subring S in O/I such that R is the preimage of S in O.
This note is a reaction to Conant and Kruckman's recent preprint `Three surprising instance of dividing'. It mainly consists of an erratum to the author's paper `Forking, imaginaries and other features of ACFG', in light of the results of the aforementioned paper.
Given a structure $\mathcal{M}$ and a stably embedded $\emptyset$-definable set $Q$, we prove tameness preservation results when enriching the induced structure on $Q$ by some further structure $\mathcal{Q}$. In particular, we show that if $T=\text{Th}(\mathcal{M})$ and $\text{Th}(\mathcal{Q})$ are stable (resp., superstable, $ω$-stable), then so is the theory $T[\mathcal{Q}]$ of the enrichment of $\mathcal{M}$ by $\mathcal{Q}$. Assuming simplicity of $T$, elimination of hyperimaginaries and a further condition on $Q$ related to the behavior of algebraic closure, we also show that simplicity and NSOP$_1$ pass from $\text{Th}(\mathcal{Q})$ to $T[\mathcal{Q}]$. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of $(\mathbb{Z},+)$. More generally, we show that any stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) countable graph can be defined in a stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) expansion of $(\mathbb{Z},+)$ by some unary predicate $A\subseteq\mathbb{N}$.
These are the notes of a course for the summer school Model Theory in Bilbao hosted by the Basque Center for Applied Mathematics (BCAM) and the Universidad del País Vasco/Euskal Herriko Unibertsitatea in September 2023. The goal of this course is to prove the Ax-Kochen-Ershov (AKE) theorem. This classical result in model theory was proven by Ax and Kochen and independently by Ershov in 1965-1966. The AKE theorem is considered as the starting point of the model theory of valued fields and witnessed numerous refinements and extensions. To a certain measure, motivic integration can be considered as such. The AKE theorem is not only an important result in model theory, it yields a striking application to $p$-adic arithmetics. Artin conjectured that all $p$-adic fields are $C_2$ (every homogeneous polynomial of degree $d$ and in $>d^2$ variable has a non trivial zero). A consequence of the AKE theorem is that the $p$-adics are asymptotically $C_2$. The conjecture of Artin has been disproved by Terjanian in 1966, yielding that the solution given by the AKE theorem is in a sense optimal. The proof presented here is due to Pas but the general strategy stays faithful to the original paper of Ax and Kochen, which consist in the study of the asymptotic first-order theory of the $p$-adics.
We study expansions of a vector space $V$ over a field $\mathbb F$, possibly with extra structure, with a generic submodule over a subring of $\mathbb F$. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP$_1$, NTP$_2$ and NSOP$_1$. We also study induced independence relations in the expansion.
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$_2$.
This paper is concerned with the model-theoretic study of pairs $(K,F)$ where $K$ is an algebraically closed field and $F$ is a distinguished subfield of $K$ allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP$_1$, and NIP. As an application, we conclude that a PAC field is NSOP$_1$ iff its absolute Galois group is (as a profinite group).
Milliet asks the following question: given two prime numbers $p\neq q$, is there a division algebra of characteristic $p$ which is of dp-rank $q^2$ and of dimension $q^2$ over its center? We answer in the affirmative. We also give an example of a finite burden central division algebra over some ultraproduct of $p$-adic numbers. As a conclusion we revisit an example of Albert to prove that there exists non-cyclic division algebras of finite dp-rank.