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Locally definable homotopy

2009-05-08
Elías Baro , Margarita Otero
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ON O-MINIMAL HOMOTOPY GROUPS

2009-03-17
Elías Baro , Margarita Otero
Journal Article ON O-MINIMAL HOMOTOPY GROUPS Get access Elías Baro, Elías Baro † Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain †Corresponding author. E-mail: [email protected] Search for other … Journal Article ON O-MINIMAL HOMOTOPY GROUPS Get access Elías Baro, Elías Baro † Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain †Corresponding author. E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero Margarita Otero Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 61, Issue 3, September 2010, Pages 275–289, https://doi.org/10.1093/qmath/hap011 Published: 17 March 2009 Article history Received: 15 August 2008 Revision received: 28 January 2009 Published: 17 March 2009
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Commutators in groups definable in o-minimal structures

2012-02-27
Elías Baro , Éric Jaligot , Margarita Otero
We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and … We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
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Approximation on Nash sets with monomial singularities

2014-05-28
Elías Baro , José F. Fernando , Jesús M. Ruíz
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Corrigendum to “Locally definable groups in o-minimal structures” [J. Algebra 301 (2006) 194–223]

2008-08-01
Elías Baro , Mário J. Edmundo
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Normal triangulations in o-minimal structures

2010-01-25
Elías Baro
Abstract Let be an o-minimal structure over a real closed field R . Given a simplicial complex K and some definable subsets S 1 , …, S l of its … Abstract Let be an o-minimal structure over a real closed field R . Given a simplicial complex K and some definable subsets S 1 , …, S l of its realization ∣ K ∣ in R we prove that there exist a subdivision K ′ of K and a definable triangulation φ′: ∣ K ′∣ → ∣ K ∣ of ∣ K ∣ partitioning S 1 , …, S l with φ ′ definably homotopic to id ∣ K ∣ . As an application of this result we obtain the semialgebraic Hauptvermutung.
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Cartan subgroups of groups definable in o-minimal structures

2013-11-28
Elías Baro , Éric Jaligot , Margarita Otero
We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the … We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely in terms of the dimension.
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on the o-minimal LS-category

2011-09-29
Elías Baro
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An Extension Result for Maps Admitting an Algebraic Addition Theorem

2018-02-17
Elías Baro , J. De Vicente , Margarita Otero
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Locally $$\mathbb {C}$$ C -Nash groups

2018-09-19
Elías Baro , J. De Vicente , Margarita Otero
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Normal triangulations in o-minimal structures

2007-01-01
Elías Baro
We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and a finite number of definable subsets of its realization |K| in … We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and a finite number of definable subsets of its realization |K| in R we prove that there exists a triangulation (K',f) of |K| compatible with the definable subsets such that K' is a subdivision of K and f is definably homotopic to the identity on |K|.

On o-minimal homotopy groups

2008-01-01
Elías Baro , Margarita Otero
We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any … We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with the study of semialgebraic homotopy done by H. Delfs and M. Knebusch allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.

Topology of definable abelian groups in o-minimal structures

2011-11-17
Elías Baro , Alessandro Berarducci
In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to … In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.
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Open core and small groups in dense pairs of topological structures

2020-07-16
Elías Baro , Amador Martín-Pizarro
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Two-dimensional locally $\mathbb{K}$-Nash groups

2017-01-01
Elías Baro , J. De Vicente , Margarita Otero
We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé's description of meromorphic maps admitting an Algebraic Addition Theorem and analyse the … We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé's description of meromorphic maps admitting an Algebraic Addition Theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification.

Locally definable and approximate subgroups of semialgebraic groups

2018-12-27
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and in particular is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Rings of differentiable semialgebraic functions

2019-01-01
Elías Baro , José F. Fernando , J. M. Gamboa
In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^r$ on a semialgebraic … In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^r$ on a semialgebraic set $M\subset\mathbb{R}^m$. Denote ${\mathcal S}^0(M)$ the ring of semialgebraic functions on $M$ that admit a continuous extension to an open semialgebraic neighborhood of $M$ in $\text{cl}(M)$. This ring is the real closure of ${\mathcal S}^r(M)$. If $M$ is locally compact, the ring ${\mathcal S}^r(M)$ enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite ${\mathcal S}^r(M)$ is not real closed for $r\geq1$, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring ${\mathcal S}^0(M)$. In addition, the quotients of ${\mathcal S}^r(M)$ by its prime ideals have real closed fields of fractions, so the ring ${\mathcal S}^r(M)$ is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of ${\mathcal S}^r(M)$ and ${\mathcal S}^0(M)$ guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring ${\mathcal S}^r(M)$ is a Gelfand ring and its Krull dimension is equal to $\dim(M)$. We also show similar properties for the ring ${\mathcal S}^{r*}(M)$ of differentiable bounded semialgebraic functions. In addition, we confront the ring ${\mathcal S}^{\infty}(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^{\infty}$ with the ring ${\mathcal N}(M)$ of Nash functions on $M$.

Locally definable homotopy

2008-01-01
Elías Baro , Margarita Otero
In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to … In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.

On the o-minimal LS-category

2009-05-09
Elías 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.

The derived subgroup of linear and simply-connected o-minimal groups

2019-02-26
Elías Baro
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Locally definable subgroups of semialgebraic groups

2019-10-07
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: … We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].
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Open core and small groups in dense pairs of topological structures

2018-01-26
Elías Baro , Amador Martín-Pizarro
Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap … Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.

Corrigendum to “Approximation on Nash sets with monomial singularities” [Adv. Math. 262 (2014) 59–114]

2015-12-02
Elías Baro , José F. Fernando , Jesús M. Ruíz

Cartan subgroups and regular points of o‐minimal groups

2019-04-01
Elías Baro , Alessandro Berarducci , Margarita Otero
Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M … Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.
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The derived subgroup of linear and simply-connected o-minimal groups

2018-02-20
Elías Baro
We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the … We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the definability of the derived subgroup in case that the group is simply-connected.

Commutators in groups definable in o-minimal structures

2010-06-01
Elías Baro , Éric Jaligot , Margarita Otero
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and … We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

Approximation on Nash sets with monomial singularities

2013-07-02
Elías Baro , José F. Fernando , Jesús M. Ruíz
This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, … This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.

Locally definable subgroups of semialgebraic groups

2018-12-27
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Spectral maps associated to semialgebraic branched coverings

2020-01-01
Elías Baro , José F. Fernando , J. M. Gamboa
In this article we prove that a semialgebraic map is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map … In this article we prove that a semialgebraic map is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the ramification set and the ramification index. A crucial fact to prove the preceding result is the characterization of the prime ideals whose fibers under the previous spectral map are singletons.

Spectral maps associated to semialgebraic branched coverings

2021-01-02
Elías Baro , José F. Fernando , J. M. Gamboa
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Two-Dimensional Locally Nash Groups

2021-03-29
Elías Baro , J. De Vicente , Margarita Otero
Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse … Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $\mathbb{R}$.

Locally definable subgroups of semialgebraic groups

2018-01-01
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Σ_{i=1}^m(X-X)$ is an approximate subgroup of $G$.
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Spectral Spaces in o-minimal and other NIP theories

2022-01-01
Elías Baro , José F. Fernando , Daniel Palacín
We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum … We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the {\em honest topology}. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon's $F_M$ retraction.
View PDF Chat

The derived subgroup of linear and simply-connected o-minimal groups

2018-01-01
Elías Baro
We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the … We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the definability of the derived subgroup in case that the group is simply-connected.
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Approximation on Nash sets with monomial singularities

2013-01-01
Elías Baro , José F. Fernando , Jesús M. Ruíz
This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, … This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.
View PDF Chat

Open core and small groups in dense pairs of topological structures

2018-01-01
Elías Baro , Amador Martín-Pizarro
Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap … Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.

Ellis enveloping semigroups in real closed fields

2023-01-01
Elías Baro , Daniel Palacín
We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of … We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgrebraic functions representing Boolean combinations of d-semialgebraic sets.
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Ellis enveloping semigroups in real closed fields

2024-03-13
Elías Baro , Daniel Palacín
Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the … Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d -semialgebraic sets.
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Rings of differentiable semialgebraic functions

2024-07-19
Elías Baro , José F. Fernando , J. M. Gamboa

Two-dimensional simply connected abelian locally Nash groups

2015-01-01
Elías Baro , J. De Vicente , Margarita Otero
The aim of this paper is to give a description of simply connected abelian locally Nash groups of dimension $2$. Along the way we prove that, for any $n\geq 2$, … The aim of this paper is to give a description of simply connected abelian locally Nash groups of dimension $2$. Along the way we prove that, for any $n\geq 2$, a locally Nash structure over $(\mathbb{R}^n,+)$ can be characterized via a meromorphic map admitting an algebraic addition theorem.

Rings of differentiable semialgebraic functions

2024-07-19
Elías Baro , José F. Fernando , J. M. Gamboa

Ellis enveloping semigroups in real closed fields

2024-03-13
Elías Baro , Daniel Palacín
Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the … Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d -semialgebraic sets.
View PDF Chat

Ellis enveloping semigroups in real closed fields

2023-01-01
Elías Baro , Daniel Palacín
We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of … We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgrebraic functions representing Boolean combinations of d-semialgebraic sets.
View PDF Chat

Spectral Spaces in o-minimal and other NIP theories

2022-01-01
Elías Baro , José F. Fernando , Daniel Palacín
We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum … We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the {\em honest topology}. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon's $F_M$ retraction.
View PDF Chat

Two-Dimensional Locally Nash Groups

2021-03-29
Elías Baro , J. De Vicente , Margarita Otero
Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse … Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $\mathbb{R}$.

Spectral maps associated to semialgebraic branched coverings

2021-01-02
Elías Baro , José F. Fernando , J. M. Gamboa
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Open core and small groups in dense pairs of topological structures

2020-07-16
Elías Baro , Amador Martín-Pizarro
View PDF Chat

Spectral maps associated to semialgebraic branched coverings

2020-01-01
Elías Baro , José F. Fernando , J. M. Gamboa
In this article we prove that a semialgebraic map is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map … In this article we prove that a semialgebraic map is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the ramification set and the ramification index. A crucial fact to prove the preceding result is the characterization of the prime ideals whose fibers under the previous spectral map are singletons.

Locally definable subgroups of semialgebraic groups

2019-10-07
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: … We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].
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Cartan subgroups and regular points of o‐minimal groups

2019-04-01
Elías Baro , Alessandro Berarducci , Margarita Otero
Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M … Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.
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The derived subgroup of linear and simply-connected o-minimal groups

2019-02-26
Elías Baro
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Rings of differentiable semialgebraic functions

2019-01-01
Elías Baro , José F. Fernando , J. M. Gamboa
In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^r$ on a semialgebraic … In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^r$ on a semialgebraic set $M\subset\mathbb{R}^m$. Denote ${\mathcal S}^0(M)$ the ring of semialgebraic functions on $M$ that admit a continuous extension to an open semialgebraic neighborhood of $M$ in $\text{cl}(M)$. This ring is the real closure of ${\mathcal S}^r(M)$. If $M$ is locally compact, the ring ${\mathcal S}^r(M)$ enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite ${\mathcal S}^r(M)$ is not real closed for $r\geq1$, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring ${\mathcal S}^0(M)$. In addition, the quotients of ${\mathcal S}^r(M)$ by its prime ideals have real closed fields of fractions, so the ring ${\mathcal S}^r(M)$ is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of ${\mathcal S}^r(M)$ and ${\mathcal S}^0(M)$ guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring ${\mathcal S}^r(M)$ is a Gelfand ring and its Krull dimension is equal to $\dim(M)$. We also show similar properties for the ring ${\mathcal S}^{r*}(M)$ of differentiable bounded semialgebraic functions. In addition, we confront the ring ${\mathcal S}^{\infty}(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^{\infty}$ with the ring ${\mathcal N}(M)$ of Nash functions on $M$.

Locally definable and approximate subgroups of semialgebraic groups

2018-12-27
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and in particular is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Locally definable subgroups of semialgebraic groups

2018-12-27
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Locally $$\mathbb {C}$$ C -Nash groups

2018-09-19
Elías Baro , J. De Vicente , Margarita Otero
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The derived subgroup of linear and simply-connected o-minimal groups

2018-02-20
Elías Baro
We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the … We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the definability of the derived subgroup in case that the group is simply-connected.

An Extension Result for Maps Admitting an Algebraic Addition Theorem

2018-02-17
Elías Baro , J. De Vicente , Margarita Otero
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Open core and small groups in dense pairs of topological structures

2018-01-26
Elías Baro , Amador Martín-Pizarro
Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap … Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.

Locally definable subgroups of semialgebraic groups

2018-01-01
Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Σ_{i=1}^m(X-X)$ is an approximate subgroup of $G$.
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The derived subgroup of linear and simply-connected o-minimal groups

2018-01-01
Elías Baro
We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the … We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the definability of the derived subgroup in case that the group is simply-connected.
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Open core and small groups in dense pairs of topological structures

2018-01-01
Elías Baro , Amador Martín-Pizarro
Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap … Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.

Two-dimensional locally $\mathbb{K}$-Nash groups

2017-01-01
Elías Baro , J. De Vicente , Margarita Otero
We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé's description of meromorphic maps admitting an Algebraic Addition Theorem and analyse the … We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé's description of meromorphic maps admitting an Algebraic Addition Theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification.

Corrigendum to “Approximation on Nash sets with monomial singularities” [Adv. Math. 262 (2014) 59–114]

2015-12-02
Elías Baro , José F. Fernando , Jesús M. Ruíz

Two-dimensional simply connected abelian locally Nash groups

2015-01-01
Elías Baro , J. De Vicente , Margarita Otero
The aim of this paper is to give a description of simply connected abelian locally Nash groups of dimension $2$. Along the way we prove that, for any $n\geq 2$, … The aim of this paper is to give a description of simply connected abelian locally Nash groups of dimension $2$. Along the way we prove that, for any $n\geq 2$, a locally Nash structure over $(\mathbb{R}^n,+)$ can be characterized via a meromorphic map admitting an algebraic addition theorem.

Approximation on Nash sets with monomial singularities

2014-05-28
Elías Baro , José F. Fernando , Jesús M. Ruíz
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Cartan subgroups of groups definable in o-minimal structures

2013-11-28
Elías Baro , Éric Jaligot , Margarita Otero
We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the … We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely in terms of the dimension.
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Approximation on Nash sets with monomial singularities

2013-07-02
Elías Baro , José F. Fernando , Jesús M. Ruíz
This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, … This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.

Approximation on Nash sets with monomial singularities

2013-01-01
Elías Baro , José F. Fernando , Jesús M. Ruíz
This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, … This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.
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Commutators in groups definable in o-minimal structures

2012-02-27
Elías Baro , Éric Jaligot , Margarita Otero
We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and … We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
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Topology of definable abelian groups in o-minimal structures

2011-11-17
Elías Baro , Alessandro Berarducci
In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to … In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.
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on the o-minimal LS-category

2011-09-29
Elías Baro
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Commutators in groups definable in o-minimal structures

2010-06-01
Elías Baro , Éric Jaligot , Margarita Otero
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and … We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

Normal triangulations in o-minimal structures

2010-01-25
Elías Baro
Abstract Let be an o-minimal structure over a real closed field R . Given a simplicial complex K and some definable subsets S 1 , …, S l of its … Abstract Let be an o-minimal structure over a real closed field R . Given a simplicial complex K and some definable subsets S 1 , …, S l of its realization ∣ K ∣ in R we prove that there exist a subdivision K ′ of K and a definable triangulation φ′: ∣ K ′∣ → ∣ K ∣ of ∣ K ∣ partitioning S 1 , …, S l with φ ′ definably homotopic to id ∣ K ∣ . As an application of this result we obtain the semialgebraic Hauptvermutung.
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On the o-minimal LS-category

2009-05-09
Elías 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.

Locally definable homotopy

2009-05-08
Elías Baro , Margarita Otero
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ON O-MINIMAL HOMOTOPY GROUPS

2009-03-17
Elías Baro , Margarita Otero
Journal Article ON O-MINIMAL HOMOTOPY GROUPS Get access Elías Baro, Elías Baro † Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain †Corresponding author. E-mail: [email protected] Search for other … Journal Article ON O-MINIMAL HOMOTOPY GROUPS Get access Elías Baro, Elías Baro † Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain †Corresponding author. E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero Margarita Otero Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 61, Issue 3, September 2010, Pages 275–289, https://doi.org/10.1093/qmath/hap011 Published: 17 March 2009 Article history Received: 15 August 2008 Revision received: 28 January 2009 Published: 17 March 2009
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Corrigendum to “Locally definable groups in o-minimal structures” [J. Algebra 301 (2006) 194–223]

2008-08-01
Elías Baro , Mário J. Edmundo
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Locally definable homotopy

2008-01-01
Elías Baro , Margarita Otero
In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to … In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.

On o-minimal homotopy groups

2008-01-01
Elías Baro , Margarita Otero
We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any … We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with the study of semialgebraic homotopy done by H. Delfs and M. Knebusch allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.

Normal triangulations in o-minimal structures

2007-01-01
Elías Baro
We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and a finite number of definable subsets of its realization |K| in … We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and a finite number of definable subsets of its realization |K| in R we prove that there exists a triangulation (K',f) of |K| compatible with the definable subsets such that K' is a subdivision of K and f is definably homotopic to the identity on |K|.
Coauthor Papers Together
Margarita Otero 13
José F. Fernando 9
J. De Vicente 5
Pantelis E. Eleftheriou 4
Jesús M. Ruíz 4
J. M. Gamboa 4
Ya’acov Peterzil 4
Éric Jaligot 3
Daniel Palacín 3
Amador Martín-Pizarro 3
Alessandro Berarducci 2
Mário J. Edmundo 1

Locally Semialgebraic Spaces

1985-01-01
Hans Delfs , Manfred Knebusch
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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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Finiteness problems on Nash manifolds and Nash sets

2014-02-08
José F. Fernando , J. M. Gamboa , Jesús M. Ruíz
We study here several finiteness problems concerning affine Nash manifolds M and Nash subsets X . Three main results are: (i) A Nash function on a semialgebraic subset Z of … We study here several finiteness problems concerning affine Nash manifolds M and Nash subsets X . Three main results are: (i) A Nash function on a semialgebraic subset Z of M has a Nash extension to an open semialgebraic neighborhood of Z in M , (ii) A Nash set X that has only normal crossings in M can be covered by finitely many open semialgebraic sets U equipped with Nash diffeomorphisms (u_1,\dots,u_m):U\to\mathbb R^m such that U\cap X=\{u_1\cdots u_r=0\} , (iii) Every affine Nash manifold with corners N is a closed subset of an affine Nash manifold M where the Nash closure of the boundary \partial N of N has only normal crossings and N can be covered with finitely many open semialgebraic sets U such that each intersection N\cap U=\{u_1\ge0,\dots u_r\ge0\} for a Nash diffeomorphism (u_1,\dots,u_m):U\to\mathbb R^m .
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A survey on groups definable in o-minimal structures

2008-05-22
Margarita Otero
Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable … Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following:
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Real Algebraic Geometry

1998-01-01
Frédéric Mangolte , Jean-Philippe Rolin , Kurdyka Krzysztof +2 more

Linear Groups Definable in o-Minimal Structures

2002-01-01
Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

o-MINIMAL FUNDAMENTAL GROUP, HOMOLOGY AND MANIFOLDS

2002-04-01
Alessandro Berarducci , Margarita Otero
The definable fundamental group of a definable set in an o-minimal expansion of a field is computed. This is achieved by proving the relevant case of the o-minimal van Kampen … The definable fundamental group of a definable set in an o-minimal expansion of a field is computed. This is achieved by proving the relevant case of the o-minimal van Kampen theorem. This result is applied to show that if the geometrical realization of a simplicial complex over an o-minimal expansion of a field is a definable manifold of dimension not 4, then its geometrical realization over the reals is a topological manifold.

On groups and fields definable in o-minimal structures

1988-09-01
Anand Pillay

Real algebraic geometry

2014-01-21
Jacek Bochnak , Michel Coste , Marie-Françoise Roy
1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- … 1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

One-dimensional Nash groups

1992-06-01
James J. Madden , Charles S. Stanton
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On torsion-free groups in o-minimal structures

2005-10-01
Ya’acov Peterzil , Sergei Starchenko
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably … We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .
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Groups definable in local fields and pseudo-finite fields

1994-02-01
Ehud Hrushovski , Anand Pillay

ON O-MINIMAL HOMOTOPY GROUPS

2009-03-17
Elías Baro , Margarita Otero
Journal Article ON O-MINIMAL HOMOTOPY GROUPS Get access Elías Baro, Elías Baro † Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain †Corresponding author. E-mail: [email protected] Search for other … Journal Article ON O-MINIMAL HOMOTOPY GROUPS Get access Elías Baro, Elías Baro † Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain †Corresponding author. E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Margarita Otero Margarita Otero Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 61, Issue 3, September 2010, Pages 275–289, https://doi.org/10.1093/qmath/hap011 Published: 17 March 2009 Article history Received: 15 August 2008 Revision received: 28 January 2009 Published: 17 March 2009
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Definably simple groups in o-minimal structures

2000-02-24
Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

Analytic Functions of Several Complex Variables.

1968-04-01
J. W. Gray , Robert C. Gunning , Hugo Rossi
The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others … The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248)

2000-05-01
A. J. Wilkie
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).

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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Locally definable homotopy

2009-05-08
Elías Baro , Margarita Otero
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A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22
Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

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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On Łojasiewiczʼs inequality and the Nullstellensatz for rings of semialgebraic functions

2013-11-04
José F. Fernando , J. M. Gamboa
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Definable homomorphisms of abelian groups in o-minimal structures

1999-11-01
Ya’acov Peterzil , Sergei Starchenko

ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

2011-09-23
José F. Fernando , J. M. Gamboa
In this work we define a semialgebraic set S ⊂ ℝ n to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral … In this work we define a semialgebraic set S ⊂ ℝ n to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01
Ya’acov Peterzil , Charles Steinhorn
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

On the size of the fibers of spectral maps induced by semialgebraic embeddings

2016-02-19
José F. Fernando
Let be the ring of (continuous) semialgebraic functions on a semialgebraic set M and its subring of bounded semialgebraic functions. In this work we compute the size of the fibers … Let be the ring of (continuous) semialgebraic functions on a semialgebraic set M and its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps and induced by the inclusion of a semialgebraic subset N of M . The ring can be understood as the localization of at the multiplicative subset of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion that reduces both problems above to an analysis of the fibers of the spectral map . If we denote , it holds that the restriction map is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of at the points of Z . The size of the fibers of prime ideals “close” to the complement provides valuable information concerning how N is immersed inside M . If N is dense in M , the map is surjective and the generic fiber of a prime ideal contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber is a finite set for . If such is the case, our procedure allows us to compute the size s of . If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in .
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On the spectra of rings of semialgebraic functions

2011-03-19
José F. Fernando , J. M. Gamboa

Super real closed rings

2007-01-01
Marcus Tressl
A super real closed ring is a commutative ring equipped with the operation of all continuous functions ${\mathbb R}^n\to {\mathbb R}$. Examples are rings of continuous functions and super real … A super real closed ring is a commutative ring equipped with the operation of all continuous functions ${\mathbb R}^n\to {\mathbb R}$. Examples are rings of continuous functions and super real fields attached to $z$-prime ideals in the sense of Dales and
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The basic theory of real closed spaces

1989-01-01
Niels Schwartz
Les espaces localement semi-algebriques sont generalises par une classe d'espaces localement anneles, appeles espaces reels fermes. Les espaces de base des espaces reels fermes affines sont des spectres reels d'anneaux, … Les espaces localement semi-algebriques sont generalises par une classe d'espaces localement anneles, appeles espaces reels fermes. Les espaces de base des espaces reels fermes affines sont des spectres reels d'anneaux, les faisceaux de structure sont appeles faisceaux fermes reels. Avec ces espaces on peut developper une theorie semblable a la theorie des schemas. Il y a un foncteur naturel de la categorie des espaces semi-algebriques a la categorie des espaces reels fermes. Par ce foncteur, on peut comparer des proprietes des espaces semi-algebriques et leurs espaces reels fermes correspondant

Higher homotopy of groups definable in o-minimal structures

2010-10-31
Alessandro Berarducci , Marcello Mamino , Margarita Otero
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Locally definable groups in o-minimal structures

2005-05-18
Mário J. Edmundo
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On the Krull dimension of rings of continuous semialgebraic functions

2015-10-29
José F. Fernando , J. M. Gamboa
Let R be a real closed field, {\mathcal S}(M) the ring of continuous semialgebraic functions on a semialgebraic set M\subset R^m and {\mathcal S}^*(M) its subring of continuous semialgebraic functions … Let R be a real closed field, {\mathcal S}(M) the ring of continuous semialgebraic functions on a semialgebraic set M\subset R^m and {\mathcal S}^*(M) its subring of continuous semialgebraic functions that are bounded with respect to R . In this work we introduce semialgebraic pseudo-compactifications of M and the semi algebraic depth of a prime ideal \mathfrak p of {\mathcal S}(M) in order to provide an elementary proof of the finiteness of the Krull dimensions of the rings {\mathcal S}(M) and {\mathcal S}^*(M) for an arbitrary semialgebraic set M . We are inspired by the classical way to compute the dimension of the ring of polynomial functions on a complex algebraic set without involving the sophisticated machinery of real spectra. We show \mathrm {dim}({\mathcal S}(M))=\mathrm {dim}({\mathcal S}^*(M))=\mathrm {dim}(M) and prove that in both cases the height of a maximal ideal corresponding to a point p \in M coincides with the local dimension of M at p . In case \mathfrak p is a prime z - ideal of {\mathcal S}(M) , its semialgebraic depth coincides with the transcendence degree of the real closed field \mathrm {qf}({\mathcal S}(M)/\mathfrak p) over R .
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On open and closed morphisms between semialgebraic sets

2011-08-02
José F. Fernando , J. M. Gamboa
In this work we study how open and closed semialgebraic maps between two semialgebraic sets extend, via the corresponding spectral maps, to the Zariski and maximal spectra of their respective … In this work we study how open and closed semialgebraic maps between two semialgebraic sets extend, via the corresponding spectral maps, to the Zariski and maximal spectra of their respective rings of semialgebraic and bounded semialgebraic functions.
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Epimorphic extensions and Prüfer extensions¶ of partially ordered rings

2000-07-01
Niels Schwartz

Simple algebraic and semialgebraic groups over real closed fields

2000-06-13
Ya’acov Peterzil , Anand Pillay , Sergei Starchenko
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic … We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

AFFINE NASH GROUPS OVER REAL CLOSED FIELDS

2011-12-01
Ehud Hrushovski , Anand Pillay
We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of … We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R = ℝ, this result was claimed in [5], but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not necessarily connected affine Nash groups over arbitrary real closed fields.
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Normal spectral spaces and their dimensions

1983-12-01
Michel Carral , Michel Coste

TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01
Anand Pillay
We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

Generix never gives up

2006-06-01
Éric Jaligot
Abstract We prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal … Abstract We prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal counterexample to the Genericity Conjecture.

Linear algebraic groups

1966-01-01
Armand Borel
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions. Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.

Introduction to the Theory of Analytic Spaces

1966-01-01
Raghavan Narasimhan
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Cosets, genericity, and the Weyl group

2009-06-03
Éric Jaligot
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Paires de structures O-minimales

1998-06-01
Yerzhan Baisalov , Bruno Poizat
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 universal covering homomorphism in o‐minimal expansions of groups

2007-09-26
Mário J. Edmundo , Pantelis E. Eleftheriou
Abstract Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism $ \tilde p $ : … Abstract Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism $ \tilde p $ : $ \tilde G $ → G is a locally definable covering homomorphism and π 1 ( G ) is isomorphic to the o‐minimal fundamental group π ( G ) of G defined using locally definable covering homomorphisms. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)
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Covers of groups definable in o-minimal structures

2005-01-01
Mário J. Edmundo
In this paper we develop the theory of covers for locally definable groups in o-minimal structures. In this paper we develop the theory of covers for locally definable groups in o-minimal structures.
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Groups with the minimal condition on centralizers

1979-10-01
R. M. Bryant

Commutators in groups definable in o-minimal structures

2012-02-27
Elías Baro , Éric Jaligot , Margarita Otero
We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and … We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
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O-minimal homotopy and generalized (co)homology

2013-04-01
Artur Piękosz
This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent … This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes.
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ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS

2014-07-01
José F. Fernando
Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$ . We prove that … Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$ . We prove that every $R$ -homomorphism ${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$ is essentially the evaluation homomorphism at a certain point $p\in F^{n}$ adjacent to the extended semialgebraic set $M_{F}$ . This type of result is commonly known in real algebra as a substitution lemma. In the case when $M$ is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.
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Solvable groups definable in o-minimal structures

2003-10-09
Mário J. Edmundo
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ON CHAINS OF PRIME IDEALS IN RINGS OF SEMIALGEBRAIC FUNCTIONS

2013-10-18
José F. Fernando
In this work, we study the structure of non-refinable chains of prime ideals in the (real closed) rings 𝒮(M) and 𝒮*(M) of semialgebraic and bounded semialgebraic functions on a semialgebraic … In this work, we study the structure of non-refinable chains of prime ideals in the (real closed) rings 𝒮(M) and 𝒮*(M) of semialgebraic and bounded semialgebraic functions on a semialgebraic set M⊂ℝm. We pay special attention to the prime z-ideals of 𝒮(M) and the minimal prime ideals of both rings. For the last, a decomposition of each semialgebraic set as an irredundant finite union of closed pure dimensional semialgebraic subsets plays a crucial role. We prove moreover the existence of maximal ideals in the ring 𝒮(M) of prefixed height whenever M is non-compact.