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Cherlin's Conjecture that an infinite simple group of finite Morely rank is an algebraic group over an algebraically closed field has been around for many years now, and has been …
Cherlin's Conjecture that an infinite simple group of finite Morely rank is an algebraic group over an algebraically closed field has been around for many years now, and has been the starting point for a considerable amount of research. In this survey paper we describe two approaches towards Cherlin's Conjecture, first without any stability assumption via the theory of algebraic groups and secondly via the theory of Tits buildings in the context of finite Morley rank. While the conjecture is still open, our results cover most classes of classical and algebraic groups and the (twisted) Chevalley groups.
These notes should be read with those of Zoé Chatzidakis. We report some results from Hrushovski and Pillay. The main items in this paper are
These notes should be read with those of Zoé Chatzidakis. We report some results from Hrushovski and Pillay. The main items in this paper are
We prove that no infinite field is definable in the theory of the free group.
We prove that no infinite field is definable in the theory of the free group.
We prove that no infinite field is definable in the theory of the free group
We prove that no infinite field is definable in the theory of the free group
We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: …
We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field $L$ is definably isomorphic to the group of $L$-rational points of an algebraic group defined over $L$.
We prove that any infinite group interpretable in a separably closed field <italic>F</italic> of finite Eršov-invariant is definably isomorphic to an <italic>F</italic>-algebraic group. Using this result we show that any …
We prove that any infinite group interpretable in a separably closed field <italic>F</italic> of finite Eršov-invariant is definably isomorphic to an <italic>F</italic>-algebraic group. Using this result we show that any infinite field <italic>K</italic> interpretable in a separably closed field <italic>F</italic> is itself separably closed; in particular, in the finite invariant case <italic>K</italic> is definably isomorphic to a finite extension of <italic>F</italic>.
We prove that any infinite group interpretable in a separably closed field F of finite Ersov-invariant is definably isomorphic to an F-algebraic group.Using this result we show that any infinite …
We prove that any infinite group interpretable in a separably closed field F of finite Ersov-invariant is definably isomorphic to an F-algebraic group.Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F .This paper answers a question raised by D. Marker, whose help and guidance made this work possible.I would like to thank A. Pillay for helpful discussions, and E.
For algebraically closed fields, one cannot define characteristic zero by means of a single sentence in the language of rings. However, for global fields (i.e., finite separable extensions of Q …
For algebraically closed fields, one cannot define characteristic zero by means of a single sentence in the language of rings. However, for global fields (i.e., finite separable extensions of Q or Fp[t]) characteristic zero is definable. In other words, there is a sentence in the language of rings which is true in a global field precisely when that global field has characteristic zero. In fact there are many subsets of the set of global fields which have a first-order definition and they correspond to the arithmetically definable subsets of the natural numbers. It turns out that characteristic zero is also definable for infinite finitely generated fields. The characterization of all subsets of infinite finitely generated fields is still an open problem.
Suppose F is a field of prime characteristic p and E is a finite subgroup of the additive group (F,+). Then E is an elementary abelian p-group. We consider two …
Suppose F is a field of prime characteristic p and E is a finite subgroup of the additive group (F,+). Then E is an elementary abelian p-group. We consider two such subgroups, say $E$ and $E'$, to be equivalent if there is an $\alpha\in F^*:= F\setminus\{0\}$ such that $E=\alpha E'$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.
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Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that …
Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$.
A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode "conjugacy" relations among subgroups of $S$, and which are modelled on the …
A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode "conjugacy" relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we study and classify extensions of $p$-local finite groups, and also compute the fundamental group of the classifying space of a $p$-local finite group.
A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the …
A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.
Abstract We prove that no infinite field is interpretable in the first‐order theory of non‐Abelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.
Abstract We prove that no infinite field is interpretable in the first‐order theory of non‐Abelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.
We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.
We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.
The aim of this paper is to set the foundation to separate geometric model theory from model theory. Our thesis is that it is possible to lift results from geometric …
The aim of this paper is to set the foundation to separate geometric model theory from model theory. Our thesis is that it is possible to lift results from geometric model theory to non first order logic (e.g. LWl)W). We introduce a relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This is important when one is trying to lift forking to nonelementary classes, in contexts where there exists pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2].
We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar …
We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group.
Abstract We prove an analytic version of the stable graph regularity lemma by Malliaris and Shelah (Trans. Amer. Math. Soc. 366 (2014), no. 3, 1551–1585), which applies to stable functions …
Abstract We prove an analytic version of the stable graph regularity lemma by Malliaris and Shelah (Trans. Amer. Math. Soc. 366 (2014), no. 3, 1551–1585), which applies to stable functions . Our methods involve continuous model theory and, in particular, results on the structure of local Keisler measures for stable continuous formulas. Along the way, we develop some basic tools around ultraproducts of metric structures and linear functionals on continuous formulas, and we also describe several concrete families of examples of stable functions.
Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of …
Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups. So, by a “definable f -generic” or $dfg$ group we mean a definable group in a saturated model with a global f -generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o -minimal groups in the p -adic context. In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f -generic type, etc. In this paper we will show that every definable f -generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o -minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f -generic subgroup of a definable f -generic group has finite index, and every f -generic type of a definable f -generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f -generic types coincide with almost periodic types in the p -adic case.
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We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of …
We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R = ℝ, this result was claimed in [5], but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not necessarily connected affine Nash groups over arbitrary real closed fields.
Abstract In this article we explore some properties of H -structures which are introduced in [2]. We describe a construction of H -structures based on one-dimensional asymptotic classes which preserves …
Abstract In this article we explore some properties of H -structures which are introduced in [2]. We describe a construction of H -structures based on one-dimensional asymptotic classes which preserves pseudofiniteness. That is, the H -structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU -rank one, there are no new definable groups in H -structures. This improves the corresponding result in [2].
Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A …
Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k -stable. Then there is a normal subgroup H ≤ G of index at most n , and a set Y ⊆ G , which is a union of cosets of H , such that | A △ Y | ≤ε| H |. It follows that, for any coset C of H , either | C ∩ A |≤ ε| H | or | C \ A | ≤ ε | H |. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$ .
Abstract We study the flow of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic …
Abstract We study the flow of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f‐generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f‐generic types of trigonalizable algebraic groups, and prove that every f‐generic type is almost periodic.
The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of topological dynamics, here the definable means a group …
The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of topological dynamics, here the definable means a group admits a global f-generic type which is over a small submodel. This definable is a dual concept to finitely satisfiable generic, and a useful tool to describe the analogue of torsion free o-minimal groups in the $p$-adic context.
In this paper we will show that every $f$-generic group in $\Q$ is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over $\Q$. This is analogous to the $o$-minimal context, where every connected torsion free group in $\R$ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show that every open $f$-generic subgroup of a $f$-generic group has finite index, and every $f$-generic type of a $f$-generic group is almost periodic, which gives a positive answer on the problem raised in \cite{P-Y} of whether $f$-generic types coincide with almost periodic types in the $p$-adic case.
Abstract We give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two …
Abstract We give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two definable sets: The set of balls of a given radius r < 1 contained in the valuation ring and the set of balls of a given multiplicative radius r < 1. We also show that in an algebraically closed valued field a 0-definable set is stably embedded if and only if its algebraic closure is stably embedded.
We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory …
We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).We generalize Buium's notion of an algebraic D-group to ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$for some ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$.
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an …
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic …
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably …
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . …
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . We also prove that the elimination of imaginaries is uniform in p . Using p -adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p ) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both …
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ (or any real closed field) where algebra alone determines the ordering and hence the topology of the field: In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but this will be too coarse to give a diferentiable structure. A celebrated example of how partial algebraic and topological data ( G a locally euclidean group) determines a differentiable structure ( G is a Lie group) is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason. The main result which we discuss here (see [13] for the full version) is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨ M , <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of M n , n = 1, 2, …, in some first order expansion ℳ of ⟨ M , <⟩.
We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: …
We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].
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Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse …
Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $\mathbb{R}$.
Abstract We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original …
Abstract We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original structure. This question has been investigated by Dolich, Miller and Steinhorn in the setting of ordered structures as part of their work on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.
We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the …
We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications. Difference field are fields with a distinguished automorphism σ. They were first studied by Ritt in the 1930s. A good reference for the algebraic results is [Cohn 1965]. Interest in the model theory of difference fields started at the end of the eighties, particularly during the MSRI logic year, because of two questions. The first question stemmed from the failure of Zil’ber’s conjecture: there is a strongly minimal theory extending the theory of algebraically closed fields of any given characteristic. People were looking at the possibility of finding a nondefinable automorphism σ of F p (the algebraic closure of the field Fp with p elements), such that Th(F p ,+ , · , σ) is strongly minimal. This question so far remains open. The second problem had to do with the difference fields Fq = (F p ,+ , · , φq), where q is a power of p and φq : x 7→ x is a power of the Frobenius automorphism x 7→ x. The hope was to generalise the work of Ax on finite fields to these structures, and in particular to describe the theory of the non-principal ultraproducts of the difference fields Fq. These questions led Macintyre, van den Dries and Wood to look for a model companion of the theory of difference fields, and to prove various results (decidability, description of the completions, etc . . . ) for this theory, henceforth called ACFA. For details and attribution of results, see [Macintyre 1997]. I should also mention that the second problem was solved recently, by Hrushovski [1996b] and Notes based on lectures given at MSRI, January 98.
Suppose G is a finite group and A\subseteq G is such that \{gA:g\in G\} has VC-dimension strictly less than k . We find algebraically well-structured sets in G which, up …
Suppose G is a finite group and A\subseteq G is such that \{gA:g\in G\} has VC-dimension strictly less than k . We find algebraically well-structured sets in G which, up to a chosen \epsilon>0 , describe the structure of A and behave regularly with respect to translates of A . For the subclass of groups with uniformly fixed finite exponent r , these algebraic objects are normal subgroups with index bounded in terms of k , r , and \epsilon . For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model-theoretic methods related to the work of Breuillard, Green, and Tao [8] and Hrushovski [28] on approximate groups, as well as a result of Alekseev, Glebskiĭ, and Gordon [1] on approximate homomorphisms.
We study the induced structure on definable groups in existentially closed difference fields. If $G$ is a definable subgroup of a semi-abelian variety, orthogonal to every definable field, we show …
We study the induced structure on definable groups in existentially closed difference fields. If $G$ is a definable subgroup of a semi-abelian variety, orthogonal to every definable field, we show that $G$ is stable and stably embedded; every definable subset of $G^n$ is a Boolean combination of cosets of definable subgroups of $G^n$, and $G^n$ has at most countably many definable subgroups. This generalises to positive characteristic earlier results of the authors.
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the …
(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the notion of a theory with built-in Skolem functions, and for a treatment of the example just mentioned. This property of Peano arithmetic obviously comes from the fact that in each nonempty definable subset of a model we can definably choose an element, namely, its least member. (1.2) Consider now a real closed field R and a nonempty subset D of R which is definable (with parameters) in R. Again we can definably choose an element of D , as follows: D is a union of finitely many singletons and intervals ( a, b ) where – ∞ ≤ a < b ≤ + ∞; if D has a least element we choose that element; if not, D contains an interval ( a, b ) for which a ∈ R ∪ { − ∞}is minimal; for this a we choose b ∈ R ∪ {∞} maximal such that ( a, b ) ⊂ D . Four cases have to be distinguished: (i) a = − ∞ and b = + ∞; then we choose 0; (ii) a = − ∞ and b ∈ R ; then we choose b − 1; (iii) a ∈ R and b ∈ = + ∞; then we choose a + 1; (iv) a ∈ R and b ∈ R ; then we choose the midpoint ( a + b )/2. It follows as in the case of Peano arithmetic that the theory RCF of real closed fields has a definitional extension with built-in Skolem functions.
The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically …
The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy. In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p -adic fields. We shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields. We want to describe the definable subsets of p -adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K ). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p -adically closed fields. If K is p -adically closed, what are the definable subsets of K ? To the best of our knowledge, this question has not been answered until now. What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p -adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. …
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.
In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]: Let V ⊂ R m be a real algebraic set, and let U …
In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]: Let V ⊂ R m be a real algebraic set, and let U ⊂ R m be an open set defined by finitely many polynomial inequalities: Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure ( U ∩ V )) then there exists a real analytic curve with p (0) = 0 and with p(t) ∈ U ∩ V for t > 0. Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V ), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps R n +1 → R n . If, in this tiny extension of Milnor's result, we replace ‘ R ’ everywhere by ‘ Q p ’, we obtain a p -adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p -adic context, may be defined just as they are over the reals: namely, as those sets obtained from p -adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.
Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M …
Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M ) equivalence relations on M n . In particular, we show that if E is an A -definable equivalence relation on M n ( A ⊂ M ) then E has only finitely many classes with nonempty interior in M n , each such class being moreover also A -definable. As a consequence, we are able to give some conditions under which an O -minimal theory T eliminates imaginaries (in the sense of Poizat [P]). If L is a first order language and M an L -structure, then by a definable set in M , we mean something of the form X ⊂ M n , n ≥ 1, where X = {( a 1 …, a n ) ∈ M n : M ⊨ ϕ (ā)} for some formula ∈ L ( M ). (Here L ( M ) means L together with names for the elements of M .) If the parameters from come from a subset A of M , we say that X is A-definable . M is said to be O-minimal if M = ( M , <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals ( a, b ) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th( M ) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M .
Here we consider some problems concerning regular types. In the first place we consider a strongly minimal set D . One can ask what is the strength of the assumption …
Here we consider some problems concerning regular types. In the first place we consider a strongly minimal set D . One can ask what is the strength of the assumption that D has (full) elimination of imaginaries (namely, every definable set X over D has as canonical parameter some tuple from D ). We show that D cannot be locally modular. Nontriviality of D is immediate. However, to exclude the locally modular nontrivial case one has to understand structures of the form G/E , where G is a modular strongly minimal group and E is a definable equivalence relation on G with finite classes. We show that the quotient structure G/E can be obtained in two steps. First quotient by a finite subgroup K of G to obtain a strongly minimal group H . Now let Γ be a finite subgroup of the group Aff( H ) of definable affine automorphisms of H (namely maps of the form x → α x + a , where α is a definable automorphism of H and a ∈ H ), and quotient H by Γ (namely form the orbit space of H under Γ). It can clearly be arranged that Γ contains no nontrivial subgroup of translations. In the second place we look at a nontrivial modular regular type p whose pregeometry is actually a geometry. The geometry is then known to be (infinite-dimensional) projective geometry over a division ring F . We ask whether F is definable (internally to p ). If F is finite, this is clear. In fact in this case p must have U -rank 1. So we assume F to be infinite. We are only able to show definability of F in the case where F is a field, using some results on 2-transitive subgroups of PGL [V]. Moreover in the superstable case we also observe that p is isolated.
This paper is about various ways in which groups arise or are of interest in model theory. In Section 1.1, I briefly introduce three important classes of first-order theories: stable …
This paper is about various ways in which groups arise or are of interest in model theory. In Section 1.1, I briefly introduce three important classes of first-order theories: stable theories, simple theories, and NIP theories. Section 1.2 is about the classification of groups definable in specific theories or structures, mainly fields, and the relationship to algebraic groups. In Section 1.3, I study generalized stability and definable groups in more detail, giving the theory of “generic types” in the various contexts. I also discuss 1-based theories and groups. Section 1.4 is about the compact Hausdorff group G/G00 attached to a definable group and how it may carry information in various contexts (including approximate subgroups and arithmetic regularity). In Section 1.5, I discuss Galois theory, including the various Galois groups attached to first-order theories, various kinds of strong types, and definable groups of automorphisms. In Section 1.6, I study various points of interaction between topological dynamics and definable groups, in particular “Newelski’s conjecture” relating G/G00 to the “Ellis group”. And in Section 1.7, I touch on the model theory of the free group.
In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]: Let V ⊂ R m be a real algebraic set, and let U …
In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]: Let V ⊂ R m be a real algebraic set, and let U ⊂ R m be an open set defined by finitely many polynomial inequalities: Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure ( U ∩ V )) then there exists a real analytic curve with p (0) = 0 and with p(t) ∈ U ∩ V for t > 0. Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V ), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps R n +1 → R n . If, in this tiny extension of Milnor's result, we replace ‘ R ’ everywhere by ‘ Q p ’, we obtain a p -adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p -adic context, may be defined just as they are over the reals: namely, as those sets obtained from p -adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.