Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the …
Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.
Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the …
Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered.
As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.
Types in o-minimal theories by Janak Daniel Ramakrishnan Doctor of Philosophy in Mathematics University of California, Berkeley Professor Thomas Scanlon, Chair We extend previous work on classifying o-minimal types, and …
Types in o-minimal theories by Janak Daniel Ramakrishnan Doctor of Philosophy in Mathematics University of California, Berkeley Professor Thomas Scanlon, Chair We extend previous work on classifying o-minimal types, and develop several applications. Marker developed a dichotomy of o-minimal types into “cuts” and “noncuts,” with a further dichotomy of cuts being either “uniquely” or “non-uniquely realizable.” We use this classification to extend work by van den Dries and Miller on bounding growth rates of definable functions in Chapter 3, and work by Marker on constructing certain “small” extensions in Chapter 4. We further sub-classify “non-uniquely realizable cuts” into three categories in Chapter 2, and we give define the notion of a “decreasing” type in Chapter 5, which is a presentation of a type well-suited for our work. Using this definition, we achieve two results: in Chapter 5.2, we improve a characterization of definable types in o-minimal theories given by Marker and Steinhorn, and in Chapter 6 we answer a question of Speissegger’s about extending a continuous function to the boundary of its domain. As well, in Chapter 5.3, we show how every elementary extension can be presented as decreasing.
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is …
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α , if T has the maximum possible number of models of size α , i.e. 2 α , then no structure theorem is expected (cf. [Sh1]). O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω -categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is …
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α , if T has the maximum possible number of models of size α , i.e. 2 α , then no structure theorem is expected (cf. [Sh1]). O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω -categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.
Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.
Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.
We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The …
We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The main tool is a theory for intersection of curves which we develop.
Abstract In this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions …
Abstract In this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions of manifolds involved in comparison withthe general case.
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ …
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of $T^*$ interprets a model of $T'$. If $T'$ is NIP, $T^*$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ …
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of $T^*$ interprets a model of $T'$. If $T'$ is NIP, $T^*$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.
Here we prove that weakly o-minimal theories of nite convexity rank having less than 2 ω countable models are almost ωcategorical.
Here we prove that weakly o-minimal theories of nite convexity rank having less than 2 ω countable models are almost ωcategorical.
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.
In this paper we study (strongly) locally o-minimal structures.We first give a characterization of the strong local o-minimality.We also investigate locally o-minimal expansions of (R, +, <).
In this paper we study (strongly) locally o-minimal structures.We first give a characterization of the strong local o-minimality.We also investigate locally o-minimal expansions of (R, +, <).
We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories.
We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories.
We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The …
We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The main tool is a theory for intersection of curves which we develop.
Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples …
Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal M can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses 'geometric calculus' which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.
The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. …
The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set.
We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly …
We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.
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 We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of …
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.
We present the details of a model-theoretic proof of an analogue of the Manin–Mumford conjecture for semiabelian varieties in positive characteristic. As a by-product of the proof we reduce the …
We present the details of a model-theoretic proof of an analogue of the Manin–Mumford conjecture for semiabelian varieties in positive characteristic. As a by-product of the proof we reduce the general positive-characteristic Mordell–Lang problem to a question about purely inseparable points on subvarieties of semiabelian varieties.
We fix an arbitrary o -minimal structure ( R , ω, …), where ( R , <) is a dense linearly ordered set without end points. In this paper “definable” …
We fix an arbitrary o -minimal structure ( R , ω, …), where ( R , <) is a dense linearly ordered set without end points. In this paper “definable” means “definable with parameters from R ”, We equip R with the interval topology and R n with the induced product topology. The main result of this paper is the following. Theorem. Let V ⊆ R n be a definable open set and suppose that f : V → R n is a continuous injective definable map. Then f is open, that is, f(U) is open whenever U is an open subset of V . Woerheide [6] proved the above theorem for o -minimal expansions of a real closed field using ideas of homology. The case of an arbitrary o -minimal structure remained an open problem, see [4] and [1]. In this paper we will give an elementary proof of the general case. Basic definitions and notation . A box B ⊆ R n is a Cartesian product of n definable open intervals: B = ( a 1 , b 1 ) × … × ( a n , b n ) for some a i , b i , ∈ R ∪ {−∞, +∞}, with a i < b i , Given A ⊆ R n , cl( A ) denotes the closure of A , int( A ) denotes the interior of A , bd( A ) ≔ cl( A ) − int( A) denotes the boundary of A , and ∂ A ≔ cl( A ) − A denotes the frontier of A , Finally, we let π: R n → R n − denote the projection map onto the first n − 1 coordinates. Background material . Without mention we will use notions and facts discussed in [5] and [3]. We will also make use of the following result, which appears in [2].
The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is …
The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion. Consider some well-known examples of ℵ 0 -stable theories: vector spaces over Q , algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case. What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p .
(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 …
(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.
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>.
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment …
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.
We prove the following two theorems on embedded o-minimal structures: Theorem 1. Let ℳ ≺ 풩 be o-minimal structures and let ℳ* be the expansion of ℳ by all traces …
We prove the following two theorems on embedded o-minimal structures: Theorem 1. Let ℳ ≺ 풩 be o-minimal structures and let ℳ* be the expansion of ℳ by all traces in M of 1-variable formulas in 풩, that is all sets of the form φ(M, ā) for ā ⊆ N and φ(x, ȳ) ∈ ℒ(풩). Then, for any N-formula ψ(x1, …, xk), the set ψ(Mk) is ℳ*-definable. Theorem 2. Let 풩 be an ω1-saturated structure and let S be a sort in 풩eq. Let 풮 be the 풩-induced structure on S and assume that 풮 is o-minimal. Then 풮 is stably embedded.