Returning to semi-bounded sets

Authors

Ya’acov Peterzil

Type: Article

Publication Date: 2009-06-01

Citations: 35

DOI: https://doi.org/10.2178/jsl/1243948329

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Abstract

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.

Locations

Journal of Symbolic Logic
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On semibounded expansions of ordered groups

2020-01-01

Pantelis E. Eleftheriou , Alex Savatovsky

We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle R, <, +, \dots\rangle$ … We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle R, <, +, \dots\rangle$ is a semibounded o-minimal structure and $P\subseteq R$ a set satisfying certain tameness conditions, then $\langle \cal R, P\rangle$ remains semibounded. Examples include the cases when $\mathcal{R}=\langle \mathbb R,<,+, (x\mapsto \lambda x)_{\lambda \in \mathbb R}, \cdot_{ [0, 1]^2} \rangle$, and $P= 2^\mathbb Z$ or $P$ is an iteration sequence. As an application, we obtain that smooth functions definable in such $\langle \mathcal R, P\rangle$ are definable in $\mathcal R$.

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Defining new linear functions in tame expansions of the real ordered additive group

2021-01-01

Alex Savatovsky

We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an … We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{R}=( \mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{R}. Examples of such structures that does define new total linear functions include the cases when $\mathcal{R}$ is a reduct of $(\mathbb{R},<,+,\cdot_{\upharpoonright (0,1)^2},(x\mapsto \lambda x)_{\lambda\in I\subseteq \mathbb{R}})$, and $P= 2^\mathbb{Z}$, or $P$ is an iteration sequence (for any $I$) or $P=\mathbb{Z}$, for $I=\mathbb{Q}$.

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Defining new linear functions in tame expansions of the real ordered additive group

2021-10-24

Alex Savatovsky

On semibounded expansions of ordered groups

2023-01-01

Pantelis E. Eleftheriou , Alex Savatovsky

We explore semibounded expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle \mathbb R, \lt , … We explore semibounded expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle \mathbb R, \lt , +, \ldots \rangle $ is a semibounded o-minimal structure and

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Definable groups as homomorphic images of semi-linear and field-definable groups

2012-03-23

Pantelis E. Eleftheriou , Ya’acov Peterzil

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Almost o-minimal structures and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="fraktur">X</mml:mi></mml:math>-structures

2022-06-01

Masato Fujita

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Definable groups as homomorphic images of semilinear and field-definable groups

2012-01-01

Pantelis E. Eleftheriou , Ya’acov Peterzil

We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves … We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their locally definable covers. As a corollary, we prove the Compact Domination Conjecture in o-minimal expansions of ordered groups.

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Definable groups as homomorphic images of semilinear and field-definable groups

2012-01-27

Pantelis E. Eleftheriou , Ya’acov Peterzil

Definable group extensions in semi‐bounded o‐minimal structures

2009-11-17

Mário J. Edmundo , Pantelis E. Eleftheriou

Abstract In this note we show: Let R = 〈 R , <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a … Abstract In this note we show: Let R = 〈 R , <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈 R m , +〉 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Invariance results for definable extensions of groups

2010-07-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

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Almost o-minimal structures and $\mathfrak X$-structures

2021-04-03

Masato Fujita

We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a … We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\mathfrak X$-sets and $\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\mathfrak X$-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded $\mathfrak X$-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups $\mathcal M=(M,<,0,+,\ldots)$. Let $\{A_\lambda\}_{\lambda\in\Lambda}$ be a finite family of definable subsets of $M^{m+n}$. Take an arbitrary positive element $R \in M$ and set $B=]-R,R[^n$. Then, there exists a finite partition into definable sets \begin{equation*} M^m \times B = X_1 \cup \ldots \cup X_k \end{equation*} such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and either $X_i \cap A_\lambda = \emptyset$ or $X_i \subseteq A_\lambda$ for any $1 \leq i \leq k$ and $\lambda \in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b \in M^m$. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.

Coverings by open cells

2013-01-01

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells. We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.

Coverings by open cells

2013-03-13

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

Definable Tietze extension property in o-minimal expansions of ordered groups

2023-05-15

Masato Fujita

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Expansion of a semi-bounded o-minimal structure by a geometric progression

2023-01-01

Masato Fujita

We demonstrate that an expansion of a semi-bounded o-minimal expansion of the ordered group of reals by an increasing geometric progression is locally o-minimal. We demonstrate that an expansion of a semi-bounded o-minimal expansion of the ordered group of reals by an increasing geometric progression is locally o-minimal.

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Structure theorems in tame expansions of o-minimal structures by a dense set

2020-08-01

Pantelis E. Eleftheriou , Ayhan Günaydın , Philipp Hieronymi

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On the o-minimal LS-category

2009-01-01

Elias Baro

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the … We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay's conjecture.

On the o-minimal LS-category

2009-05-09

Elías Baro

Definable Tietze extension property in o-minimal expansion of ordered group

2020-11-03

Masato Fujita

The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any … The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $f:A \rightarrow M$ defined on a definable closed subset of $M^n$ has a definable continuous extension $F:M^n \rightarrow M$.

Structures having o-minimal open core

2009-10-08

Alfred Dolich , Chris Miller , Charles Steinhorn

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. … The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: <disp-quote> <italic>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an expansion of a densely ordered group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma asterisk right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>∗</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,>,*)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is definably complete and satisfies the uniform finiteness property. Then the open core of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is o-minimal.</italic> </disp-quote> Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.

On Pillay's conjecture in the general case

2017-03-17

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

Expansions of real closed fields that introduce no new smooth functions

2020-03-23

Pantelis E. Eleftheriou , Alex Savatovsky

On semibounded expansions of ordered groups

2023-01-01

Pantelis E. Eleftheriou , Alex Savatovsky

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Lie-like decompositions of groups definable in o-minimal structures

2009-01-01

Annalisa Conversano

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study … There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.

Interpretable groups are definable

2014-03-18

Pantelis E. Eleftheriou , Ya’acov Peterzil , Janak Ramakrishnan

We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product … We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.

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Pillay's conjecture and its solution—a survey

2010-06-07

Ya’acov Peterzil

Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the … Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups.

Definable quotients of locally definable groups

2012-03-23

Pantelis E. Eleftheriou , Ya’acov Peterzil

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Interpretable groups are definable

2011-10-30

Janak Ramakrishnan , Ya’acov Peterzil , Pantelis E. Eleftheriou

Definable groups as homomorphic images of semi-linear and field-definable groups

2012-03-23

Pantelis E. Eleftheriou , Ya’acov Peterzil

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On central extensions and definably compact groups in o-minimal structures

2010-11-17

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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Definable quotients of locally definable groups

2011-03-24

Pantelis E. Eleftheriou , Ya’acov Peterzil

We study locally definable abelian groups $\CU$ in various settings and examine conditions under which the quotient of $\CU$ by a discrete subgroup might be definable. This turns out to … We study locally definable abelian groups $\CU$ in various settings and examine conditions under which the quotient of $\CU$ by a discrete subgroup might be definable. This turns out to be related to the existence of the type-definable subgroup $\CU^{00}$ and to the divisibility of $\CU$.

On central extensions and definably compact groups in o-minimal structures

2008-01-01

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple … We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).

Additive reducts of real closed fields and strongly bounded structures

2023-10-21

Hind Abu Saleh , Ya’acov Peterzil

Additive reducts of real closed fields and strongly bounded structures Additive reducts of real closed fields and strongly bounded structures

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Definable groups as homomorphic images of semilinear and field-definable groups

2012-01-27

Pantelis E. Eleftheriou , Ya’acov Peterzil

Definable structures in o-minimal theories: One dimensional types

2010-10-31

Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

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A note on generic subsets of definable groups

2011-01-01

Mário J. Edmundo , Giuseppina Terzo

We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably … We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.

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Pillay's conjecture for groups definable in weakly o‐minimal non‐valuational structures

2021-05-04

Pantelis E. Eleftheriou

Let G be a group definable in a weakly o-minimal non-valuational structure M. Then G / G 00 , equipped with the logic topology, is a compact Lie group, and … Let G be a group definable in a weakly o-minimal non-valuational structure M. Then G / G 00 , equipped with the logic topology, is a compact Lie group, and if G has finitely satisfiable generics, then dim ( G / G 00 ) = dim ( G ) . Our main technical result is that G is a dense subgroup of a group definable in the canonical o-minimal extension of M.

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Compact domination for groups definable in linear o-minimal structures

2009-07-15

Pantelis E. Eleftheriou

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Groups definable in o-minimal structures: various properties and a diagram

2020-01-01

Annalisa Conversano

We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and … We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.

Groups definable in weakly o-minimal non-valuational structures

2020-01-01

Pantelis E. Eleftheriou

Let $\mathcal M$ be a weakly o-minimal non-valuational structure, and $\mathcal N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $\mathcal M$ is a … Let $\mathcal M$ be a weakly o-minimal non-valuational structure, and $\mathcal N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $\mathcal M$ is a subgroup of a group $K$ definable in $\mathcal N$, which is canonical in the sense that it is the smallest such group. As an application, we obtain that $G^{00}= G\cap K^{00}$, and establish Pillay's Conjecture in this setting: $G/G^{00}$, equipped with the logic topology, is a compact Lie group, and if $G$ has finitely satisfiable generics, then $\dim_{Lie}(G/G^{00})= \dim(G)$.

Definable group extensions in semi‐bounded o‐minimal structures

2009-11-17

Mário J. Edmundo , Pantelis E. Eleftheriou

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Non-standard lattices and o-minimal groups

2013-03-01

Pantelis E. Eleftheriou

Abstract We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice . We … Abstract We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice . We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.

On the o-minimal Hilbert's fifth problem

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal … Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k>0$, the $k$-torsion subgroups of $G;$ the o-minimal cohomology algebra over ${\mathbb Q}$ of $G.$ As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

Coverings by open cells

2014-01-28

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

Coverings by open cells

2013-03-13

Mário J. Edmundo , Pantelis E. Eleftheriou , Luca Prelli

Expansions of real closed fields which introduce no new smooth functions

2018-01-01

Pantelis E. Eleftheriou , Alex Savatovsky

We prove the following theorem: let $\widetilde{\mathcal R}$ be an expansion of the real field $\overline{\mathbb R}$, such that every definable set (I) is a uniform countable union of semialgebraic … We prove the following theorem: let $\widetilde{\mathcal R}$ be an expansion of the real field $\overline{\mathbb R}$, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a "semialgebraic chunk". Then every definable smooth function $f:X\subseteq \mathbb R^n\to \mathbb R$ with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions $\widetilde{\mathcal R} = \langle \overline{\mathbb R}, P\rangle$ of the real field, such as when $P=2^\mathbb Z$, or $P\subseteq \mathbb R$ is an iteration sequence. A generalization of the theorem to d-minimal expansions $\widetilde{\mathcal R}$ of $\mathbb R_{an}$ fails. On the other hand, we prove our theorem for expansions$\widetilde{\mathcal R}$ of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as $\langle \overline{\mathbb R}, \mathbb R_{alg}, 2^\mathbb Z\rangle$.

Large groups are o-minimal

2018-05-29

Pantelis E. Eleftheriou

Let $\mathcal N$ be an expansion of an o-minimal structure $\cal M$ such that every open definable set is definable in $\mathcal M$. Assume that $\cal N$ admits a dimension … Let $\mathcal N$ be an expansion of an o-minimal structure $\cal M$ such that every open definable set is definable in $\mathcal M$. Assume that $\cal N$ admits a dimension function compatible with $\mathcal M$. Among all definable groups, we characterize those definable in $\mathcal M$ as the ones that have maximal dimension; namely, their dimension equals the dimension of their topological closure. Our setting includes all known tame expansions $\mathcal N=\langle \mathcal M, P\rangle$, where $P\subseteq M$ is a dense set. As an application of the above characterization, we obtain that if $P$ is independent, then every definable group is already definable in $\mathcal M$.

Local analysis for semi-bounded groups

2012-01-01

Pantelis E. Eleftheriou

An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called semi-bounded if it does not expand a real closed field. Possibly, it defines a real … An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called semi-bounded if it does not expand a real closed field. Possibly, it defines a real closed field with bounded domain $I\subseteq M$. Let us call

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CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES

2019-07-16

Pantelis E. Eleftheriou

Abstract We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good … Abstract We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.

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MAXIMAL COMPACT SUBGROUPS IN THE O-MINIMAL SETTING

2013-04-11

Annalisa Conversano

A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role … A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role of maximal tori is played by maximal 0-subgroups. Along the way we give structural theorems for solvable groups, linear groups, and extensions of definably compact by torsion-free definable groups.

On the Euler characteristic of definable groups

2011-02-01

Mário J. Edmundo

We show that in an arbitrary o-minimal structure the following are equivalent: (i) conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if … We show that in an arbitrary o-minimal structure the following are equivalent: (i) conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if the o-minimal Euler characteristic of the quotient is non zero; (ii) every infinite, definably connected, definably compact definable group has a non trivial torsion point (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Generic stability, forking, and þ-forking

2012-07-24

Darío García , Alf Onshuus , Alexander Usvyatsov

Abstract notions of “smallness” are among the most important tools that model theory offers for the analysis of arbitrary structures. The two most useful notions of this kind are forking … Abstract notions of “smallness” are among the most important tools that model theory offers for the analysis of arbitrary structures. The two most useful notions of this kind are forking (which is closely related to certain measure zero ideals) and thorn-forking (which generalizes the usual topological dimension). Under certain mild assumptions, forking is the finest notion of smallness, whereas thorn-forking is the coarsest. In this paper we study forking and thorn-forking, restricting ourselves to the class of generically stable types. Our main conclusion is that in this context these two notions coincide. We explore some applications of this equivalence.

Defining new linear functions in tame expansions of the real ordered additive group

2021-10-24

Alex Savatovsky

Invariance results for definable extensions of groups

2010-07-01

Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield

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The six Grothendieck operations on o-minimal sheaves

2014-01-04

Mário J. Edmundo , Luca Prelli

In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) … In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) Kunneth formula; (v) local and global Verdier duality.

A growth dichotomy for o-minimal expansions of ordered groups

1998-01-01

Chris Miller , Sergei Starchenko

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an o-minimal expansion of a divisible ordered abelian group … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an o-minimal expansion of a divisible ordered abelian group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma plus comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,>,+,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a distinguished positive element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then the following dichotomy holds: Either there is a <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>-definable binary operation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dot"> <mml:semantics> <mml:mo>⋅</mml:mo> <mml:annotation encoding="application/x-tex">\cdot</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma plus comma dot comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,>,+,\cdot ,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an ordered real closed field; or, for every definable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper R right-arrow upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:R\to R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a <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>-definable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda element-of StartSet 0 EndSet union upper A u t left-parenthesis upper R comma plus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>∪</mml:mo> <mml:mi>Aut</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \in \{0\}\cup \operatorname {Aut}(R,+)</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="limit Underscript x right-arrow plus normal infinity Endscripts left-bracket f left-parenthesis x right-parenthesis minus lamda left-parenthesis x right-parenthesis right-bracket element-of upper R"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:munder> <mml:mo stretchy="false">[</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>λ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M colon equals left-parenthesis upper M comma greater-than comma ellipsis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:mo>:=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {M}:=(M,>,\dots )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable groups with underlying set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Generic sets in definably compact groups

2007-01-01

Ya’acov Peterzil , Anand Pillay

A subset $X$ of a group $G$ is called left generic if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called left generic if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i

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On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996-01-01

Margarita Otero , Ya’acov Peterzil , Anand Pillay

Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in … Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.

Compact domination for groups definable in linear o-minimal structures

2009-07-15

Pantelis E. Eleftheriou

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On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

A trichotomy theorem for o-minimal structures

1998-11-01

Ya’acov Peterzil , Sergei Starchenko

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.

Linear O-minimal structures

1993-02-01

James Loveys , Ya’acov Peterzil

Structure theorems for o-minimal expansions of groups

2000-03-01

Mário J. Edmundo

Groups definable in ordered vector spaces over ordered division rings

2007-12-01

Pantelis E. Eleftheriou , Sergei Starchenko

Abstract Let M = 〈 M , +, <, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 … Abstract Let M = 〈 M , +, <, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 G , ⊕, e G 〉 an n -dimensional group definable in M . We show that if G is definably compact and definably connected with respect to the t -topology, then it is definably isomorphic to a ‘definable quotient group’ U/L , for some convex V -definable subgroup U of 〈 M n , +〉 and a lattice L of rank n . As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L .

Definable group extensions in semi‐bounded o‐minimal structures

2009-11-17

Mário J. Edmundo , Pantelis E. Eleftheriou

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Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01

Mário J. Edmundo , Margarita Otero

Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the … Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.

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A structure theorem for semibounded sets in the reals

1992-09-01

Ya’acov Peterzil

In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, <› that properly expands ℳ = ‹ℝ, +, <, λ a › a ∈ℝ , … In [MPP] it was shown that in every reduct of = ‹ ℝ, +, ·, <› that properly expands ℳ = ‹ℝ, +, <, λ a › a ∈ℝ , all the bounded semi-algebraic (that is, -definable) sets are definable. Said differently, every such is an expansion of = ‹ℝ, +, <, λ a , B i › a ∈ℝ, i ∈ I where { B i } i ∈ I is the collection of all bounded semialgebraic sets and the λ a 's are scalar multiplication by a . In [PSS] (see Theorem 1.2 below) it was shown that the structure is a proper reduct of ; that is, one cannot define in it all the semialgebraic sets. In [Pe] we show that is the only reduct properly between ℳ and . As a first step towards this result, we investigate in this paper the definable sets in reducts such as . (We point out that ‘definable’ will always mean ‘definable with parameters’.) Definition 1.1. Let X ⊆ ℝ n . X is called semi-bounded if it is definable in the structure ‹ℝ, +, <, λ a , B 1 , …, B k › a ∈ℝ , where the B i 's are bounded subsets of ℝ n . The main result of this paper (see Theorem 3.1) shows roughly that, in Ominimal expansions of that satisfy the partition condition (see Definition 2.3), every semibounded set can be partitioned into finitely many sets, each of which is of a form similar to a cylinder. Namely, these sets are obtained through the “stretching” of a bounded cell by finitely many linear vectors. As a corollary (see Theorem 1.4), we get different characterizations of semibounded sets, either in terms of their structure or in terms of their definability power. The following result, by A. Pillay, P. Scowcroft and C. Steinhorn, was the main motivation for this paper. The theorem is formulated here in a slightly stronger form than originally, but the proof itself is essentially the original one. A short version of the proof is included in §4.

Tame Topology and O-minimal Structures

1998-05-07

L. P. D. van den Dries

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.

o-MINIMAL COHOMOLOGY: FINITENESS AND INVARIANCE RESULTS

2009-12-01

Alessandro Berarducci , Antongiulio Fornasiero

The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the … The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language.

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o-Minimal Structures

1996-06-20

Lou van den Dries

Abstract The theory of o-minimal structures is a creation of model theorists, but is also intimately connected to “tame topology”. By tame topology I refer to the emerging idea that … Abstract The theory of o-minimal structures is a creation of model theorists, but is also intimately connected to “tame topology”. By tame topology I refer to the emerging idea that spaces of “finite type” deserve special attention: these spaces have aspects that tend to be overlooked when treated only within the general framework of (differential) topology. On the other hand, these aspects become prominent when we view these spaces as definable in model-theoretically “nice” structures. Thus the theory of o-minimal structures realizes to a large extent Grothendieck’s program of ‘topologie moderee’ outlined on pp. 25-42 of the fascinating document [Grothendieck, 1984). In some ways it also goes beyond this program.