Abstract Let = ⟨ M , +, <, 0, S ⟩ be a linear o-minimal expansion of an ordered group, and G = ⟨ G , ⊕, e G ) …
Abstract Let = ⟨ M , +, <, 0, S ⟩ be a linear o-minimal expansion of an ordered group, and G = ⟨ G , ⊕, e G ) an n -dimensional group definable in . We show that if G is definably connected with respect to the t -topology, then it is definably isomorphic to a definable quotient group U / L. for some convex ∨-definable subgroup U of ⟨ M n , +⟩ and a lattice L of rank equal to the dimension of the ‘compact part’ of G .
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in …
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in M n that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some M k with the topology induced by M k . Part of this result holds for the wider class of so-called invariant groups: each invariant group G in M n has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as M n .
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.
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable …
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable …
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.
We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) …
We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical euclidean topology on definable sets, definable order topologies, definable quotient spaces and definable metric spaces. We use o-minimality to undertake their study in topological terms, focussing here in particular on spaces of dimension one. We present several results, given in terms of piecewise decompositions and existence of definable embeddings and homeomorphisms, for various classes of spaces that are described in terms of classical separation axioms and definable analogues of properties such as separability, compactness and metrizability. For example, we prove that all Hausdorff one-dimensional definable topologies are piecewise the euclidean, discrete, or upper or lower limit topology; we give a characterization of all one-dimensional, regular, Hausdorff definable topologies in terms of spaces that have a lexicographic ordering or a topology generalizing the Alexandrov double of the euclidean topology; and we show that, if the underlying structure expands an ordered field, then any one-dimensional Hausdorff definable topology that is piecewise euclidean is definably homeomorphic to a euclidean space. As applications of these results, we prove definable versions of several open conjectures from set-theoretic topology, due to Gruenhage and Fremlin, on the existence of a 3-element basis for regular, Hausdorff topologies and on the nature of perfectly normal, compact, Hausdorff spaces; we obtain universality results for some classes of Hausdorff and regular topologies; and we characterize when certain metrizable definable topologies admit a definable metric.
Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M . We prove that the quotient $G/{\cal …
Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M . We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K , which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$ . It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$ ), and homotopy equivalent to K . This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.
Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and …
Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 \le r <\infty$. We prove that the set of definable Morse functions is open and dense in the set of definable $C^r$ functions on $X$ with respect to the definable $C^2$ topology.
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.
We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank …
We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings.
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find …
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω -categorical geometric theory interprets an infinite vector space over a finite field.
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.
We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types …
We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite rank. We prove that such theories are coordinatised by thorn-minimal types and that such a type is unstable if an only if every non-algebraic extension thereof is. We conclude that a type is stable if and only if it admits a coordinatisation in thorn-minimal stable types. We also show that non-trivial thorn-minimal stable types extend stable sets.
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.
Abstract An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an …
Abstract An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an expansion of the group by bounded predicates and group automorphisms). It is shown that every such structure has an elementary extension N such that either N is a reduct of an ordered vector space, or there is an o-minimal structure , with the same universe but of different language from N , with (i) Every definable set in N is definable in , and (ii) has an elementary substructure in which every bounded interval admits a definable real closed field. As a result certain questions about definably compact groups can be reduced to either ordered vector spaces or expansions of real closed fields. Using the known results in these two settings, the number of torsion points in definably compact abelian groups in expansions of ordered groups is given. Pillay's Conjecture for such groups follows.
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.
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 .
Abstract It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group …
Abstract It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o -minimal structure is totally transcendental of finite Morley rank.
Abstract An ordered structure is o -minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are …
Abstract An ordered structure is o -minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are ominimal. All theories considered will be o -minimal. A theory is said to be n-ary if every formula is equivalent to a Boolean combination of formulas in n free variables. (A 2-ary theory is called binary .) We prove that if a theory is not binary then it is not rc-ary for any n . We also characterize the binary theories which have a Dedekind complete model and those whose underlying set order is dense. In [5], it is shown that if T is a binary theory, is a Dedekind complete model of T , and I is an interval in , then for all cardinals K there is a Dedekind complete elementary extension of , so that . In contrast, we show that if T is not binary and is a Dedekind complete model of T , then there is an interval I in so that if is a Dedekind complete elementary extension of .