(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of having proved that theorem.
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).
Lou van den Dries. Tame topology and o-minimal structures. London Mathematical Society lecture note series, no. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, x + 180 …
Lou van den Dries. Tame topology and o-minimal structures. London Mathematical Society lecture note series, no. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, x + 180 pp. - Volume 6 Issue 2
Apparue au debut des annees 80 a la frontiere entre la geometrie reelle et la theorie des modeles, la theorie des structures o-minimales est, sans aucun doute, la realisation la …
Apparue au debut des annees 80 a la frontiere entre la geometrie reelle et la theorie des modeles, la theorie des structures o-minimales est, sans aucun doute, la realisation la plus avancee du programme de geometrie moderee propose par GROTHENDIECK dans son Esquisse d'un Programme. Depuis, l'activite des specialistes des structures o-minimales se divise principalement en deux : obtenir des resultats generaux valables dans toutes les structures o-minimales d'une part, etablir la o-minimalite de certaines structures geometriques et construire de nouveaux exemples de structures o-minimales d'autre part. C'est dans le cadre de cette seconde problematique que se situe ce travail : pour construire une nouvelle structure o-minimale, est-il necessaire de construire de nouvelles fonctions d'une seule variable; plus precisement, deux structures o-minimales peuvent-elles definir les memes ensembles d'arite deux sans pour autant definir les memes ensembles en toute arite. Dans cette these, nous donnons une reponse positive a cette question. Mieux, dans sa seconde partie, nous prouvons que la structure des ensembles sous-analytiques globaux ℝan et la structure ℝan(n) des ensembles sous-analytiques globaux definis a l'aide des fonctions analytiques restreintes de n variables definissent les memes ensembles d'arite n + 1 mais que la premiere definit strictement plus d'ensembles que la seconde en arite n + 2. Dans la premiere partie, nous decrivons les structures o-minimales C∞-lisses qui definissent, en arite deux, tous les sous-ensembles de ℝ2 semi-algebriques et eux seuls. Nous prouvons qu'il y a exactement deux telles structures: la structure engendree par les courbes semi-algebriques et la structure de tous les ensembles semi-algebriques.
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G …
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].
The study of embedded minimal surfaces in ℝ3 is a classical problem, dating to the mid 1700s, and many people have made key contributions. We will survey a few recent …
The study of embedded minimal surfaces in ℝ3 is a classical problem, dating to the mid 1700s, and many people have made key contributions. We will survey a few recent advances, focusing on joint work with Tobias H. Colding of MIT and Courant Institute, and taking the opportunity to focus on results that have not been highlighted elsewhere.
Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by …
Abstract We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
Abstract We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are …
Abstract We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are that (or rather its theory) is “rosy”. P has NIP in T and that P is stably 1-embedded in T . This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T . Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]).
Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that three tameness conditions hold. We prove that …
Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that three tameness conditions hold. We prove that the induced structure on $P$ by $\mathcal M$ eliminates imaginaries. As a corollary, we obtain that every small set $X$ definable in $\widetilde{\mathcal M}$ can be definably embedded into some $P^l$, uniformly in parameters, settling a question from [10]. We verify the tameness conditions in three examples: dense pairs of real closed fields, expansions of $\mathcal M$ by a dense independent set, and expansions by a dense divisible multiplicative group with the Mann property. Along the way, we point out a gap in the proof of a relevant elimination of imaginaries result in Wencel [17]. The above results are in contrast to recent literature, as it is known in general that $\widetilde{\mathcal M}$ does not eliminate imaginaries, and neither it nor the induced structure on $P$ admits definable Skolem functions.
Building on the positive solution of Pillay’s conjecture we present a notion of “intrinsic” reduction for elliptic curves over a real closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> …
Building on the positive solution of Pillay’s conjecture we present a notion of “intrinsic” reduction for elliptic curves over a real closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We compare such a notion with the traditional algebro-geometric reduction and produce a classification of the group of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-points of an elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with three “real” roots according to the way <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduces (algebro-geometrically) and the geometric complexity of the “intrinsically” reduced curve.
Building on the positive solution of Pillay's conjecture we present a notion of intrinsic reduction for elliptic curves over a closed field K. We compare such notion with the traditional …
Building on the positive solution of Pillay's conjecture we present a notion of intrinsic reduction for elliptic curves over a closed field K. We compare such notion with the traditional algebro-geometric reduction and produce a classification of the group of K-points of an elliptic curve E with three real roots according to the way E reduces (algebro-geometrically) and the geometric complexity of the intrinsically reduced curve.
We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the `geometric' sorts which suffice to code all imaginaries in …
We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the `geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.
Abstract We let R be an o-minimal expansion of a field, V a convex subring, and ( R 0 , V 0 ) an elementary substructure of ( R , …
Abstract We let R be an o-minimal expansion of a field, V a convex subring, and ( R 0 , V 0 ) an elementary substructure of ( R , V ). Our main result is that ( R , V ) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R , provided that k R (the residue field with structure induced from R ) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k with structure induced from ( R , V ). We also give a criterion for a superstructure of ( R , V ) being an elementary extension of ( R , V ).
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of …
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).
In this thesis is presented a study of groups of the form G/G^{00}, where G is a 1-dimensional, definably compact, definably connected, definable group in a saturated real closed field …
In this thesis is presented a study of groups of the form G/G^{00}, where G is a 1-dimensional, definably compact, definably connected, definable group in a saturated real closed field M, with respect to a notion called 1-basedness.
In particular G will be one of the following:
1. ([-1,1),+ mod 2)
2. ([1/b,b),*mod b^2
3. (SO_2(M)*) and truncations
4. (E(M)^0,+) and truncations, where E is an elliptic curve over M,
where a truncation of a linearly or circularly ordered group (G,*) is a group whose underlying set is an interval [a,b) containing the identity of G, and whose operation is *mod(b*a^{-1}).
Such groups G/G^{00} are only hyperdefinable, i.e., quotients of a definable group by a type-definable equivalence relation, in M, and therefore we consider a
suitable expansion M' in which G/G^{00} becomes definable.
We obtain that M' is interdefinable with a real closed valued field M_w, and that 1-basedness of G/G^{00} is related to the internality of G/G^{00} to either the
residue field or the value group of M_w.
In the case when G is the semialgebraic connected component of the M-points of an elliptic curve E, there is a relation between the internality of G/G^{00} to the residue field or the value group of M_w and the notion of algebraic geometric
reduction. Among our results is the following:
If G = E(M)^0, the expansion of M by a predicate for G^{00} is interdefinable with a real closed valued field M_w and G/G^{00} is internal to the value group of M_w if and only if E has split multiplicative reduction; G/G^{00} is internal to the residue field of M_w if and only if E has good reduction or nonsplit multiplicative reduction.
Abstract We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of …
Abstract We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are …
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures …
A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.
Let L be a first order language containing a binary relation symbol <. Definition. Suppose ℳ is an L -structure and < is a total ordering of the domain of …
Let L be a first order language containing a binary relation symbol <. Definition. Suppose ℳ is an L -structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal ( -minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ. In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L -theory we say that T is strongly ( - minimal if and only if every model of T is -minimal. The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω -stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2 ∣ T ∣ ) + , and characterize the strongly -minimal theories with models order isomorphic to ( R , <).
Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to …
Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U þ -rank 1 must arise from definable stable sets.
Abstract A box type is an n -type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of …
Abstract A box type is an n -type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M . From this, we deduce various structure theorems for subsets of M k , definable in the expansion of M by all convex subsets of the line. We show that after naming constants, is model complete provided M is model complete.
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G …
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].
Abstract A subset A ⊆ M of a totally ordered structure M is said to be convex , if for any a, b ∈ A : [ a < b …
Abstract A subset A ⊆ M of a totally ordered structure M is said to be convex , if for any a, b ∈ A : [ a < b → ∀ t ( a < t b → t ∈ A )]. A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some ∅-definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory T . any expansion M + of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63). that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.
Let L be a first order language. If M is an L -structure, let L M be the expansion of L obtained by adding constants for the elements of M …
Let L be a first order language. If M is an L -structure, let L M be the expansion of L obtained by adding constants for the elements of M . Definition. A type is definable if and only if for any L -formula , there is an L M -formula so that for all iff M ⊨ d θ(¯). The formula d θ is called the definition of θ. Definable types play a central role in stability theory and have also proven useful in the study of models of arithmetic. We also remark that it is well known and easy to see that for M ≺ N , the property that every M -type realized in N is definable is equivalent to N being a conservative extension of M , where Definition. If M ≺ N , we say that N is a conservative extension of M if for any n and any L N -definable S ⊂ N n , S ∩ M n is L M -definable in M . Van den Dries [Dl] studied definable types over real closed fields and proved the following result. 0.1 (van den Dries), (i) Every type over ( R , +, -,0,1) is definable . (ii) Let F and K be real closed fields and F ⊂ K. Then, the following are equivalent : (a) Every element of K that is bounded in absolute value by an element of F is infinitely close (in the sense of F) to an element of F . (b) K is a conservative extension of F .
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are …
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.