Abstract It had been conjectured that, in every countable stable theory which is categorical in some power, one can define finitely many classical geometries (cf. Example 6.2) which determine the …
Abstract It had been conjectured that, in every countable stable theory which is categorical in some power, one can define finitely many classical geometries (cf. Example 6.2) which determine the structure. More precisely, there were two conjectures.
Abstract A structure ( M , <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets …
Abstract A structure ( M , <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U -rank 1.
Abstract If K is a field of finite Morley rank, then for any parameter set A ⊆ K eq the prime model over A is equal to the model-theoretic algebraic …
Abstract If K is a field of finite Morley rank, then for any parameter set A ⊆ K eq the prime model over A is equal to the model-theoretic algebraic closure of A . A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl(∅).
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types …
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
Zusammenfassung Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen …
Zusammenfassung Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. Dies beantwortet eine alte Frage von Zilber.
Abstract An ω -categorical supersimple group is finite-by-abelian-by-finite, and has finite SU -rank. Every definable subgroup is commensurable with an acl(ø)-definable subgroup. Every finitely based regular type in a CM-trivial …
Abstract An ω -categorical supersimple group is finite-by-abelian-by-finite, and has finite SU -rank. Every definable subgroup is commensurable with an acl(ø)-definable subgroup. Every finitely based regular type in a CM-trivial ω -categorical simple theory is non-orthogonal to a type of SU -rank 1. In particular, a supersimple ω -categorical CM-trivial theory has finite SU -rank.
Abstract We define the notion of generic for an arbitrary subgroup H of a stable group, and show that H has a definable hull with the same generic properties. We …
Abstract We define the notion of generic for an arbitrary subgroup H of a stable group, and show that H has a definable hull with the same generic properties. We then apply this to the theory of stable fields.
We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined …
We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.
Abstract A type analysable in one-based types in a simple theory is itself one-based.
Abstract A type analysable in one-based types in a simple theory is itself one-based.
If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p. 2000 Mathematics Subject Classification 03C45, 03C60.
If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p. 2000 Mathematics Subject Classification 03C45, 03C60.
This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has …
This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.
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Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is …
Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal quotient. In the supersimple case, the Berline-Lascar decomposition works. One-based simple groups are finite-by-abelian-by-finite.
Abstract. 1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type …
Abstract. 1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type p ′ interbounded with p , which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ. 2. If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σ ω , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while fixing Σ generically In case p is Σ-internal and T is stable, this is the binding group of p over Σ.
Abstract We define an ℜ-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. …
Abstract We define an ℜ-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for ℜ-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are ℜ-groups.
Given a definably amenable approximate subgroup A of a (local) group in some first-order structure, there is a type-definable subgroup H normalized by A and contained in A 4 such …
Given a definably amenable approximate subgroup A of a (local) group in some first-order structure, there is a type-definable subgroup H normalized by A and contained in A 4 such that every definable superset of H has positive measure.
Abstract We develop a Sylow theory for stable groups satisfying certain additional conditions (2- finiteness, solvability or smallness) and show that their maximal p -subgroups are locally finite and conjugate. …
Abstract We develop a Sylow theory for stable groups satisfying certain additional conditions (2- finiteness, solvability or smallness) and show that their maximal p -subgroups are locally finite and conjugate. Furthermore, we generalize a theorem of Baer-Suzuki on subgroups generated by a conjugacy class of p -elements.
Abstract We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial …
Abstract We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.
We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and …
We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.
The notion of a large set in an arbitrary group is introduced in analogy to the generic sets in an algebraic or stable group. The question is studied which properties …
The notion of a large set in an arbitrary group is introduced in analogy to the generic sets in an algebraic or stable group. The question is studied which properties “satisfied largely” by a group hold on the entire group. ZUSAMMENFASSUNG. Wir definieren den Begriff einer groβen Teilmenge einer be-liebigen Gruppe in Analogie zu den generischen Mengen einer algebraischen oder stabilen Gruppe und untersuchen, welche “grofiteils erfullten” Gruppeneigenschaften fur die ganze Gruppe gelten müssen.
We show that a simple group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite Morley Rank acting faithfully on as divisible abelian group …
We show that a simple group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite Morley Rank acting faithfully on as divisible abelian group must be a linear group over some algebraically closed field.
Abstract We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low. supershort and superlow theories. An example …
Abstract We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low. supershort and superlow theories. An example of a low non supershort theory is given.
A small profinite m-stable group has an open abelian subgroup of finite $\cal M$-rank and finite exponent.
A small profinite m-stable group has an open abelian subgroup of finite $\cal M$-rank and finite exponent.
We prove for a wide class of structures that if F is a family of substructures which are pairwise uniformly commensurable, then there is a commensurable substructure invariant under automorphisms …
We prove for a wide class of structures that if F is a family of substructures which are pairwise uniformly commensurable, then there is a commensurable substructure invariant under automorphisms stabilizing F setwise. 1991 Mathematics Subject Classification 03G10, 05E20, 20E36.
A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the …
A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjectu
Abstract A minimal field of non-zero characteristic is algebraically closed.
Abstract A minimal field of non-zero characteristic is algebraically closed.
Abstract In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the …
Abstract In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries , and the study of polygroups .
Abstract We prove that a stable solvable group G which satisfies x n = 1 generically is of finite exponent dividing some power of n . Furthermore, G is nilpotent-by-finile. …
Abstract We prove that a stable solvable group G which satisfies x n = 1 generically is of finite exponent dividing some power of n . Furthermore, G is nilpotent-by-finile. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).
Abstract Non- n -ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof …
Abstract Non- n -ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.
Abstract We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability …
Abstract We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgroup of positive dimension, and a pseudofinite group of dimension 2 contains a soluble subgroup of dimension 2.
Abstract We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small stable abelian group such that no infinite definable quotient …
Abstract We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small stable abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.
Abstract We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p ′ interalgebraic …
Abstract We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p ′ interalgebraic with a finite tuple of realizations of p , which is generated over φ . Moreover, the group of elementary permutations of p ′ over all realizations of φ is type-definable.
For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and …
For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and Carbone to a representation--theoretic construction $G^{\lambda}(R)$ corresponding to an integrable representation $V^{\lambda}$ with dominant integral weight $\lambda$. When $R=\Z$, we prove that this map extends to a group homomorphism $\rho_{\lambda,\Z}: G(\Z) \to G^{\lambda}(\Z).$ We prove that the kernel $K^{\lambda}$ of the map $\rho_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group homomorphism $\varphi:G(\Z)\to G(\C)$ is injective, then $K^{\lambda}\leq H(\Z)\cong(\Z/2\Z)^{rank(G)}$.
Résumé Un groupe interprétable dans le mauvais corps vert est isogène à un quotient d’un sous-groupe définissable d’un groupe algébrique par une puissance du groupe vert. Un sous-groupe définissable d’un …
Résumé Un groupe interprétable dans le mauvais corps vert est isogène à un quotient d’un sous-groupe définissable d’un groupe algébrique par une puissance du groupe vert. Un sous-groupe définissable d’un groupe algébrique dans un corps vert ou rouge est une extension des points colorés d’un groupe algébrique multiplicatif ou additif par un groupe algébrique. En particulier, tout groupe simple définissable dans un corps coloré est algébrique.
We introduce a generalisation of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalises …
We introduce a generalisation of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalises one-basedness. We show that, under this condit
We prove certain properties of stable groups of finite exponent.In particular, an 9ΐ-group of finite exponent has normal-by-finite 2-Sylow subgroups; if it has exponent 3 2" for some n < …
We prove certain properties of stable groups of finite exponent.In particular, an 9ΐ-group of finite exponent has normal-by-finite 2-Sylow subgroups; if it has exponent 3 2" for some n < ω, then it is nilpotent-byfinite.We give an easy proof of the fact that a locally finite subgroup of a stable group of finite exponent is nilpotent-by-finite.For groups of infinite exponent, we prove the definability of an algebraically closed field of characteristic 2 under certain circumstances.Finally, we prove two general propositions about normal subgroups of stable groups.In this paper we shall be concerned with an arbitrary subgroup of a stable group (in short: a substable group).Recall that a subgroup H of a group G is definable if there is some formula φ(x) with H = <p(G); a subgroup is typedefinable if it is the intersection of definable subgroups in a saturated model.If H < G is any subgroup and K < H is such that there is a formula φ with K = φ(H), then K is relatively definable (with respect to //); the definition for relative type-definability is analogous.Note that if //is substable and K<H relatively definable, then also H/K is substable: If φ(H) = K, then H/K may be viewed as subgroup of G/φ(G); if K < H we may prefer to replace φ(x) by φ{χ) -ΛheHΨ(x h )> then N G (ψ(G)) > //and we may view H/K as subgroup of N G (ψ(G))/\l/(G).The connected component H° of a substable group H is the intersection with H of all definable subgroups K such that the index IH: H Π KI is finite; H° is normal in H and itself connected.If H is definable, we need only consider definable subgroups K < H of finite index.So the index \H:H°\ is at most 2 |7 Ί and a saturated model and for a type-definable Hthe connected component has comparable size.But in the absence of saturation or if H is just substable, H° may even be reduced to the identity!For a relatively definable H this may be remedied in some cases by considering the locally connected component // c , which is the intersection of all conjugates H 8 such that the index \H:H Π H 8 \ is finite.By Baldwin-Saxl, this is again a relatively definable subgroup of finite index.Finally, a group is small if its theory has only countably many pure types.
Abstract If G is an omega-stable group with a normal definable subgroup H , then the Sylow-2-subgroups G/H are the images of the Sylow-2-subgroups of G . Zusammenfassung. Sei G …
Abstract If G is an omega-stable group with a normal definable subgroup H , then the Sylow-2-subgroups G/H are the images of the Sylow-2-subgroups of G . Zusammenfassung. Sei G eine omega-stabile Gruppe und H ein definierbarer Normalteiler von G . Dann sind die Sylow-2-Untergruppen von G/H Bilder der Sylow-2-Untergruppen von G .
A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the …
A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the group of affine transformations of an algebraically closed field. In particular, a sharply 2-transitive permutation group of finite Morley rank of characteristic 3 is the group of affine transformations of an algebraically closed field of characteristic 3.Without any assumption on Morley rank, a sharply 2-transitive permutation group of characteristic 0 splits if its point stabilizers are virtually abelian.
In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial …
In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.
Abstract If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in X ∪ Y has a normal hyperdefinable X -internal subgroup …
Abstract If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in X ∪ Y has a normal hyperdefinable X -internal subgroup N such that G / N is Y -internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developed. Moreover, a version of Schlichting’s Theorem for hyperdefinable families of commensurable subgroups is shown.
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types …
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
We show that the action of two infinite commuting invariant rings of endomorphisms of a finite-dimensional virtually connected irreducible bi-module linearizes into a vector space over a definable field. The …
We show that the action of two infinite commuting invariant rings of endomorphisms of a finite-dimensional virtually connected irreducible bi-module linearizes into a vector space over a definable field. The same holds if the action is merely by strongly commuting endogenies, modulo some finite katakernel.
Any simple pseudofinite group G is known to be isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in …
Any simple pseudofinite group G is known to be isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that G is finite-dimensional with additive and fine dimension and, in particular, that if dim(G)=3 then G is isomorphic to PSL(2,F) for some pseudofinite field F. We describe pseudofinite finite-dimensional groups when the dimension is fine, additive and \<4 and, in particular, show that the classification G isomorphic to PSL(2,F) is independent from CFSG.
A sharply 2-transitive permutation group of characteristic 0 whose point stabiliser has an abelian subgroup of finite index splits. More generally, a near-domain of characteristic 0 with a multiplicative subgroup …
A sharply 2-transitive permutation group of characteristic 0 whose point stabiliser has an abelian subgroup of finite index splits. More generally, a near-domain of characteristic 0 with a multiplicative subgroup of finite index avoiding all multipliers $d_{a,b}$ must be a near-field. In particular this answers question 12.48 b) of the Kourovka Notebook in characteristic 0.
The focus of the conference were recent interactions between model theory, group theory and combinatorics in finite geometries. In some cases, in particular in non-archimedean geometry or combinatorics in finite …
The focus of the conference were recent interactions between model theory, group theory and combinatorics in finite geometries. In some cases, in particular in non-archimedean geometry or combinatorics in finite geometries, model theory appeared as tool. In other cases, like in ergodic theory and dynamics or in the theory of stable groups and more general neo-stable algebraic structures like valued fields, the focus was on model theoretic questions and classification results for such structures. In this way, the conference presented the broad range of topics of modern model theory.
Abstract If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable …
Abstract If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$ . This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.
We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results …
We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that pseudofinite groups contain big finite-by-abelian subgroups, and pseudofinite groups of dimension 2 contain big soluble subgroups.
Abstract We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability …
Abstract We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgroup of positive dimension, and a pseudofinite group of dimension 2 contains a soluble subgroup of dimension 2.
We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G^n is already the whole of G^n …
We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G^n is already the whole of G^n , for various equationally defined subsets. We link this with the inner measure of the set of solutions of an equation in a group, leading to new results and/or proofs in equational probabilistic group theory.
An $\omega$-categorical group of finite burden is virtually finite-by-abelian; an $\omega$-categorical ring of finite burden is virtually finite-by-null; an $\omega$-categorical NTP2 ring is virtually nilpotent.
An $\omega$-categorical group of finite burden is virtually finite-by-abelian; an $\omega$-categorical ring of finite burden is virtually finite-by-null; an $\omega$-categorical NTP2 ring is virtually nilpotent.
A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the …
A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the group of affine transformations of an algebraically closed field. In particular, a sharply 2-transitive permutation group of finite Morley rank of characteristic 3 is the group of affine transformations of an algebraically closed field of characteristic 3.Without any assumption on Morley rank, a sharply 2-transitive permutation group of characteristic 0 splits if its point stabilizers are virtually abelian.
Abstract Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these …
Abstract Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.
There is no bad group of Morley rank 2n+1 with an abelian Borel subgroup of Morley rank n. In particular, there is no bad group of Morley rank 3 (O. …
There is no bad group of Morley rank 2n+1 with an abelian Borel subgroup of Morley rank n. In particular, there is no bad group of Morley rank 3 (O. Fr{\'e}con).
For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and …
For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and Carbone to a representation--theoretic construction $G^{\lambda}(R)$ corresponding to an integrable representation $V^{\lambda}$ with dominant integral weight $\lambda$. When $R=\Z$, we prove that this map extends to a group homomorphism $\rho_{\lambda,\Z}: G(\Z) \to G^{\lambda}(\Z).$ We prove that the kernel $K^{\lambda}$ of the map $\rho_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group homomorphism $\varphi:G(\Z)\to G(\C)$ is injective, then $K^{\lambda}\leq H(\Z)\cong(\Z/2\Z)^{rank(G)}$.
If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. …
If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.
There is no bad group of Morley rank 2n+1 with an abelian Borel subgroup of Morley rank n. In particular, there is no bad group of Morley rank 3 (O. …
There is no bad group of Morley rank 2n+1 with an abelian Borel subgroup of Morley rank n. In particular, there is no bad group of Morley rank 3 (O. Fr{é}con).
This conference was about recent interactions of model theory with combinatorics, geometric group theory and the theory of valued fields, and the underlying pure model-theoretic developments. Its aim was to …
This conference was about recent interactions of model theory with combinatorics, geometric group theory and the theory of valued fields, and the underlying pure model-theoretic developments. Its aim was to report on recent results in the area, and to foster communication between the different communities.
Abstract If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in X ∪ Y has a normal hyperdefinable X -internal subgroup …
Abstract If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in X ∪ Y has a normal hyperdefinable X -internal subgroup N such that G / N is Y -internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developed. Moreover, a version of Schlichting’s Theorem for hyperdefinable families of commensurable subgroups is shown.
The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent …
The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index, and is itself of bounded index in a hyperdefinable subgroup.
Résumé Un groupe interprétable dans le mauvais corps vert est isogène à un quotient d’un sous-groupe définissable d’un groupe algébrique par une puissance du groupe vert. Un sous-groupe définissable d’un …
Résumé Un groupe interprétable dans le mauvais corps vert est isogène à un quotient d’un sous-groupe définissable d’un groupe algébrique par une puissance du groupe vert. Un sous-groupe définissable d’un groupe algébrique dans un corps vert ou rouge est une extension des points colorés d’un groupe algébrique multiplicatif ou additif par un groupe algébrique. En particulier, tout groupe simple définissable dans un corps coloré est algébrique.
There is no sad group of Morley rank 2n + 1 with an abelian Borel subgroup of rank n. In particular, Fr{é}con's Theorem follows: There is no bad group of …
There is no sad group of Morley rank 2n + 1 with an abelian Borel subgroup of rank n. In particular, Fr{é}con's Theorem follows: There is no bad group of Morely rank 3.
Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different …
Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.
We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of …
We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.
A pseudofinite group satisfying the uniform chain condition on centralizers up to finite index has a big finite-by-abelian subgroup.
A pseudofinite group satisfying the uniform chain condition on centralizers up to finite index has a big finite-by-abelian subgroup.
A bounded automorphism of a field or a group with trivial approximate centre is definable. In an expansion of a field by a Pfaffian family F of additive endomorphisms such …
A bounded automorphism of a field or a group with trivial approximate centre is definable. In an expansion of a field by a Pfaffian family F of additive endomorphisms such that algebraic closure in the expansion coincides with relative field-algebraic closure of the F-substructure generated, a bounded endomorphism, possibly composed with a power of the Frobenius, is a composition of endomorphisms associated with F.
In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable …
In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators.
Given a definably amenable approximate subgroup A of a (local) group in some first-order structure, there is a type-definable subgroup H normalized by A and contained in A 4 such …
Given a definably amenable approximate subgroup A of a (local) group in some first-order structure, there is a type-definable subgroup H normalized by A and contained in A 4 such that every definable superset of H has positive measure.
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine …
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.
Abstract We define the notion of generic for an arbitrary subgroup H of a stable group, and show that H has a definable hull with the same generic properties. We …
Abstract We define the notion of generic for an arbitrary subgroup H of a stable group, and show that H has a definable hull with the same generic properties. We then apply this to the theory of stable fields.
We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is …
We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial.
Abstract The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical …
Abstract The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.
Abstract We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic …
Abstract We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.
Abstract Let T be a countable, small simple theory. In this paper, we prove that for such T , the notion of Lascar strong type coincides with the notion of …
Abstract Let T be a countable, small simple theory. In this paper, we prove that for such T , the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.
Abstract The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed …
Abstract The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed fields. The book contains almost all the known results in the subject. Trying to attract pure group theorists in the subject and to prepare the graduate student to start the research in the area, the authors adopted an algebraic and self evident point of view rather than a model theoretic one, and developed the theory from scratch. All the necessary model theoretical and group theoretical notions are explained in length. The book is full of exercises and examples and one of its chapters contains a discussion of open problems and a program for further research.
Abstract If K is a field of finite Morley rank, then for any parameter set A ⊆ K eq the prime model over A is equal to the model-theoretic algebraic …
Abstract If K is a field of finite Morley rank, then for any parameter set A ⊆ K eq the prime model over A is equal to the model-theoretic algebraic closure of A . A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl(∅).
We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined …
We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.
Abstract We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial …
Abstract We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types …
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory …
A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.
Abstract We prove the following theorem. Let G be a connected simple bad group (i.e. of finite Morley rank, nonsolvable and with all the Borel subgroups nilpotent) of minimal Morley …
Abstract We prove the following theorem. Let G be a connected simple bad group (i.e. of finite Morley rank, nonsolvable and with all the Borel subgroups nilpotent) of minimal Morley rank. Then the Borel subgroups of G are conjugate to each other, and if B is a Borel subgroup of G , then , N G (B) = B , and G has no involutions.
Abstract We define an ℜ-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. …
Abstract We define an ℜ-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for ℜ-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are ℜ-groups.
Let T 1 and T 2 be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We …
Let T 1 and T 2 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 T 1 ∪ T 2 has a strongly minimal completion.
Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is …
Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal quotient. In the supersimple case, the Berline-Lascar decomposition works. One-based simple groups are finite-by-abelian-by-finite.
Abstract An ω -categorical supersimple group is finite-by-abelian-by-finite, and has finite SU -rank. Every definable subgroup is commensurable with an acl(ø)-definable subgroup. Every finitely based regular type in a CM-trivial …
Abstract An ω -categorical supersimple group is finite-by-abelian-by-finite, and has finite SU -rank. Every definable subgroup is commensurable with an acl(ø)-definable subgroup. Every finitely based regular type in a CM-trivial ω -categorical simple theory is non-orthogonal to a type of SU -rank 1. In particular, a supersimple ω -categorical CM-trivial theory has finite SU -rank.
Abstract We develop a Sylow theory for stable groups satisfying certain additional conditions (2- finiteness, solvability or smallness) and show that their maximal p -subgroups are locally finite and conjugate. …
Abstract We develop a Sylow theory for stable groups satisfying certain additional conditions (2- finiteness, solvability or smallness) and show that their maximal p -subgroups are locally finite and conjugate. Furthermore, we generalize a theorem of Baer-Suzuki on subgroups generated by a conjugacy class of p -elements.
This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has …
This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.
Abstract This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety …
Abstract This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.
We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem …
We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem and the Weil–Hrushovski group chunk theorem. We also study locally modular hyperdefinable groups and prove that they are bounded-by-Abelian-by-bounded. Finally, we analyze hyperdefinable groups in supersimple theories.
In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical …
In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].
A criterion is given for a strong type in a finite rank stable theory $T$ to be (almost) internal to a given nonmodular minimal type. The motivation comes from results …
A criterion is given for a strong type in a finite rank stable theory $T$ to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for a compact complex analytic space to be “algebraic” (namely Moishezon). The canonical base property for a stable theory states that the type of the canonical base of a stationary type over a realisation is almost internal to the minimal types of the theory. It is conjectured that every finite rank stable theory has the canonical base property. It is shown here, that in a theory with the canonical base property, if $p$ is a stationary type for which there exists a family of types $q_b$, each internal to a nonlocally modular minimal type $r$, and such that any pair of independent realisations of $p$ are “connected” by the $q_b$’s, then $p$ is almost internal to $r$.
These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical …
These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay \[2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler \[4] as an introduction to model theory. Furthermore, the texts of Casanovas \[1] and Wagner \[5] may also be useful to the reader to obtain a different approach to stability theory.
Dans Poizat [1981], le second auteur a montré qu'un sous-groupe infiniment définissable d'un groupe stable était intersection de sous-groupes définissables; il a posé la question de savoir si une relation …
Dans Poizat [1981], le second auteur a montré qu'un sous-groupe infiniment définissable d'un groupe stable était intersection de sous-groupes définissables; il a posé la question de savoir si une relation d'équivalence E , infiniment définissable dans un modèle M d'une théorie stable T , était conjonction de relations d'équivalence définissables. Nous allons voir ici que c'est presque exact: c'est vrai si T est totalement transcendante, et, dans le cas général de stabilité E a toujours un raffinement E 1 (plus précisément, E 1 est la conjonction de E et de la relation “ x et y ont même type”) qui a cette propriété; cela montre que cette relation E n'introduit pas d'imaginaires d'une nature vraiment différente de celle des imaginaires de Shelah: dans une théorie stable, un imaginaire infinitaire n'est rien d'autre qu'un ensemble d'imaginaires finis. La démonstration du théorème principal de cette note s'appuie lourdement sur la construction M eq de Shelah, la machinerie de la déviation, les paramètres imaginaires canoniques pour la définition d'un type stable, etc…. Pour tout cela, les références adéquates sont Shelah [1978], Pillay [1983], et Poizat [1985, Chapitre 16]. Nouscommençons par préciser ce que nous entendons par “relation d'équivalence infiniment définissable”: une collection de formules e( , ȳ ), et ȳ étant de longueur n , telle que, pour tout modèle M de T , les couples ( , ȳ ) qui les satisfont toutes forment une rélation d'équivalence E .
Abstract A type analysable in one-based types in a simple theory is itself one-based.
Abstract A type analysable in one-based types in a simple theory is itself one-based.
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math …
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
Abstract Non- n -ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof …
Abstract Non- n -ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.
On sait, depuis les travaux de Zil'ber et de Cherlin, que le degré de Morley de la théorie T d'un groupe G totalement transcendant est l'indice du plus petit sous-groupe …
On sait, depuis les travaux de Zil'ber et de Cherlin, que le degré de Morley de la théorie T d'un groupe G totalement transcendant est l'indice du plus petit sous-groupe définissable d'indice fini de G . Il est clair qu'il lui est supérieur, et l'inégalité inverse peut s'obtenir de la manière suivante: on fait agir G sur les types de S 1 ( G ) de rang de Morley maximum en associant à p , type de x au-dessus de G , le type ap de ax au-dessus de G ; on montre alors que cette action est définissable, que le fait que ap = q équivaut au fait que a satisfasse une certaine formule à paramètres dans G , ce qui est bien facile si on n'oublie pas que dans une théorie stable tous les types sont définissables; on montre ensuite que cette action est transitive, que si p et q sont de rang de Morley maximum il existe a dans G tel que ap = q , et la méthode la plus rapide, mais qui est aussi la plus sophistiquée, est d'utiliser l'argument de symétrie de la déviation employé dans la preuve de la Proposition 1 de présent article; on conclut alors puisque le degré de Morley, qui est par définition le nombre de types de rang de Morley maximum, ést egal à l'indice du stabilisateur de p , qui est définissable. Ce comportement des types de rang de Morely maximum se retrouve sans peine, si G est seulement superstable, dans celui des types de “plus petit rang continu” (encore appellé “degré de Shelah”) maximum. Pour trouver ce qui leur correspond dans le cas où G est seulement stable, il faut être un peu plus soigneux, et considérer les types p de S 1 ( G ), où G aura éventuellement été remplacé par une extension élémentaire suffisament saturée, tels que pour tout a de G ap ne dévie pas sur ∅: on montre qu'ils existent, et qu'ils sont tous conjugués par action de G ; le fait que ap = q s'exprimera cette fois par une infinité de formules et non plus par une seule.
Zusammenfassung Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen …
Zusammenfassung Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. Dies beantwortet eine alte Frage von Zilber.
We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in …
We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
Abstract We prove that a stable solvable group G which satisfies x n = 1 generically is of finite exponent dividing some power of n . Furthermore, G is nilpotent-by-finile. …
Abstract We prove that a stable solvable group G which satisfies x n = 1 generically is of finite exponent dividing some power of n . Furthermore, G is nilpotent-by-finile. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).
If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p. 2000 Mathematics Subject Classification 03C45, 03C60.
If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p. 2000 Mathematics Subject Classification 03C45, 03C60.
Abstract The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are …
Abstract The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in §3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for §3 only) differential fields [8].