arXiv subjects
Publications and source records attributed to Yatir Halevi.
For any infinite cardinal $κ$, we give an example of a centerless group $G$ of cardinality $κ$ whose automorphism tower terminates after $κ^+$ steps, answering a question of Simon Thomas. The work on this paper was assisted by AI.
For an $\aleph_0$-saturated weakly o-minimal expansion of an ordered group $\mathcal{M}$, it is shown that $\mathcal{M}^{\text{eq}}$ has uniform finiteness if and only if the collection of definable convex subgroups of $\mathcal{M}$ has uniform finiteness. If $\mathcal{M}$ expands an ordered field, then considering definable convex valuation subrings is sufficient. The results use a criterion of Johnson for uniform finiteness in $\mathcal{M}^{\text{eq}}$, [8]. In addition, it is shown that uniform finiteness in $\mathcal{M}^{\text{eq}}$ may fail for weakly o-minimal expansions of fields.
It is shown that finite-index extensions and finite-index subgroups of $ω$-stable groups can be model-theoretically wild. More precisely, there exists an $ω$-stable group $G$ such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of $G$ and in some finite-index subgroup of $G$.
We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $Γ$, and the closed $0$-balls $K/\mathcal{O}$.
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 $σ$ 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.
We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $ν(G)$, generated by the four infinitesimal subgroups $ν_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $Γ$ and $K/\mathcal{O}$. To show that $ν(G)$ is type-definable, we show that the resulting subgroups $ν_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $ν(G)$. Among others, we show that $ν(G_1\times G_2)=ν(G_1)\times ν(G_2)$ and that if $G_1\le G$ is a definable subgroup then $ν(G_1)$ is relatively definable in $ν(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(ν(G))$ and elimination of imaginaries.
Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field $\textbf{k}_F$ and valuation ring $\mathcal{O}_F$, there exists an integral separated smooth group scheme $\mathcal{E}$ over $\mathcal{O}_F$ with $\mathcal{E}\times_{\text{Spec } \mathcal{O}_F}\text{Spec } F\cong E$. If $\text{char}(\textbf{k}_F)\neq 2,3$ then one can be found over $\mathcal{O}_{F^{alg}}$ such that the definable group $\mathcal{E}(\mathcal{O})$ is the maximal generically stable subgroup of $E$. We also give some partial results on general Abelian varieties over $F$. The construction of $\mathcal{E}$ is by means of generating a birational group law over $\mathcal{O}_F$ by the aid of a generically stable generic type of a definable subgroup of $E$.
Suppose that $G$ is a graph of cardinality $μ^+$ with chromatic number $χ(G)\geq μ^+$. One possible reason that this could happen is if $G$ contains a clique of size $μ^+$. We prove that this is indeed the case when the edge relation is stable. When $G$ is a random graph (which is simple but not stable), this is not true. But still if in general the complete theory of $G$ is simple, $G$ must contain finite cliques of unbounded sizes.
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.
Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and $|\{p\text{ prime}:[G:pG]=\infty\}|<\infty$. We apply this to show that if $K$ is a strongly dependent field, then $(K,v)$ is strongly dependent for any henselian valuation $v$.
We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field $K$, its residue field $\mathbf{k}$ (when infinite), its value group $Γ$, or $K/\mathcal{O}$, where $\mathcal{O}$ is the valuation ring. Our work uses and extends techniques developed in [11] to circumvent elimination of imaginaries.
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 $A\subseteq\mathbb{N}$.
Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that $G/H$ is definably isomorphic to a $K$-linear group. The result remains true in models of $\mathrm{Th}(\mathbb{Q}_p^{an})$.
We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.
We give a geometric description of the pair $(V,p)$, where $V$ is an affine algebraic variety over a non-trivially valued algebraically closed field $K$ with valuation ring $\mathcal{O}_K$ and $p$ is a Zariski dense generically stable type concentrated on $V$, by defining a fully faithful functor to the category of schemes over $\mathcal{O}_K$ with residual dominant morphisms over $\mathcal{O}_K$. We also study a maximum modulus principle on schemes over $\mathcal{O}_K$ and show that the schemes obtained by this functor enjoy it.
Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i) $V$-minimal, (ii) power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then $\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\mathbb{Q}_p$ is definably isomorphic to a finite extension of $\mathbb{Q}_p$, answering a question of Pillay's. Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if $\mathcal{K}$ is an infinite dp-minimal pure field then every field definable in $\mathcal{K}$ is definably isomorphic to a finite extension of $K$. The proof avoids elimination of imaginaries in $\mathcal{K}$ replacing it with a reduction of the problem to certain distinguished quotients of $K$.
We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field where definable functions are generically differentiable.
We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(ω)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.