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.
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in …
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
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:
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.
Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties …
Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φ M to φ R and vice versa. Then, we apply these transfer results to give a new proof of a result of M . Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.
Abstract Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means …
Abstract Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper we present a recursive nonstandard model of normal open induction with an unbounded set of infinite prime elements.
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
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.
Abstract We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a …
Abstract We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣ f(x,y) ∣ < 1 for every (x,y) Є C , where C is the curve defined on the real closure of M by C : x 2 + y 2 = a and a > 0 is a nonstandard element of M .
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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.
Let $\cal M$ be an o-minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. We show that a definably compact definably …
Let $\cal M$ be an o-minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. We show that a definably compact definably connected group is divisible.
It is known that full Peano Arithmetic does not have the joint embedding property(JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We …
It is known that full Peano Arithmetic does not have the joint embedding property(JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We prove, using some conservation results about fragments of arithmetic, that if T is a theoryconsistent with PA and T ⊢ I E 1 − (bounded existential parameter-free induction), then any two m dels of PA which jointly embed in a model of T also jointly embed in an elementary extension of one of them. In particular, any fragment of PA extending I E 1 − fails to have JEP.
It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher …
It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible.
Abstract Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the …
Abstract Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings. Here we study the problem of representability of an element a of a model M of NOI (in some extension of M ) by a quadratic form of the type X 2 + b Y 2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M [ x, y ] with x 2 + by 2 = a can be embedded in a model of NOI. We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M . Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.
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.
A normal Z-ring Mis a discretely ordered ring, integrally closed in its fraction field and such that for each positive integer n, M/nM-Z/nZ as rings.Here we study some properties of …
A normal Z-ring Mis a discretely ordered ring, integrally closed in its fraction field and such that for each positive integer n, M/nM-Z/nZ as rings.Here we study some properties of finite generic normal Z-rings.We give a uniform universal definition of N in them.And we separate existentially closed normal Z-rings via generics.vzi,...,Znxy(x,y±o*x n + zχχ n ~ιy + ••• + z n y n = 0->aw(x= wy))and the Z-ring axioms:The theory of normal Z-rings plays a relevant role in the study of the fragment of arithmetic Normal Open Induction (NOI).NOI is the V3-theory in the language <£ which consists of NZR together with Vx((0(x,O) Λ My > O(0(x,y) -0(x,y + 1)) -Vy > O0(x,y)) for every quantifier-free <£-formula 0(x, y) (x denotes an «-tuple (x ϊ9 ... ,x n )).In [7] Shepherdson gave the following useful characterization of models of NOI:Let M be a normal discretely ordered ring.Then M is a model of NOI if and only if for every element a of the real closure of the fraction field of M there is an element amMsuch that |a -a\ < 1.From this several corollaries are deduced.Let us mention some of them.Let Mbe a model of NOI.Then every quantifier-free definable set in Mis a finite
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.
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.
Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset …
Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset of G then X contains a torsion point of G. Along the way we develop a general theory for so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.
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.
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.
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the …
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the ring of finite adeles over Q. We show that various structures associated to a prime ideal, such as quotients and localizations, are well understood model-theoretically, and they are closely connected to ultrafilters on the set of standard primes.
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 $\mathbb{R}$.
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the …
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the ring of finite ad\`eles over Q. We show that various structures associated to a prime ideal, such as quotients and localizations, are well understood model-theoretically, and they are closely connected to ultrafilters on the set of standard primes.
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.
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 $\mathbb{R}$.
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the …
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the ring of finite adeles over Q. We show that various structures associated to a prime ideal, such as quotients and localizations, are well understood model-theoretically, and they are closely connected to ultrafilters on the set of standard primes.
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the …
We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the ring of finite ad\`eles over Q. We show that various structures associated to a prime ideal, such as quotients and localizations, are well understood model-theoretically, and they are closely connected to ultrafilters on the set of standard primes.
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.
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.
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.
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.
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.
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.
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
Let $\cal M$ be an o-minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. We show that a definably compact definably …
Let $\cal M$ be an o-minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. We show that a definably compact definably connected group is divisible.
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:
It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher …
It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible.
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.
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.
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset …
Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset of G then X contains a torsion point of G. Along the way we develop a general theory for so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.
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.
Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties …
Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φ M to φ R and vice versa. Then, we apply these transfer results to give a new proof of a result of M . Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.
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.
Abstract Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means …
Abstract Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper we present a recursive nonstandard model of normal open induction with an unbounded set of infinite prime elements.
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in …
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
Abstract Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the …
Abstract Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings. Here we study the problem of representability of an element a of a model M of NOI (in some extension of M ) by a quadratic form of the type X 2 + b Y 2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M [ x, y ] with x 2 + by 2 = a can be embedded in a model of NOI. We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M . Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.
It is known that full Peano Arithmetic does not have the joint embedding property(JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We …
It is known that full Peano Arithmetic does not have the joint embedding property(JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We prove, using some conservation results about fragments of arithmetic, that if T is a theoryconsistent with PA and T ⊢ I E 1 − (bounded existential parameter-free induction), then any two m dels of PA which jointly embed in a model of T also jointly embed in an elementary extension of one of them. In particular, any fragment of PA extending I E 1 − fails to have JEP.
A normal Z-ring Mis a discretely ordered ring, integrally closed in its fraction field and such that for each positive integer n, M/nM-Z/nZ as rings.Here we study some properties of …
A normal Z-ring Mis a discretely ordered ring, integrally closed in its fraction field and such that for each positive integer n, M/nM-Z/nZ as rings.Here we study some properties of finite generic normal Z-rings.We give a uniform universal definition of N in them.And we separate existentially closed normal Z-rings via generics.vzi,...,Znxy(x,y±o*x n + zχχ n ~ιy + ••• + z n y n = 0->aw(x= wy))and the Z-ring axioms:The theory of normal Z-rings plays a relevant role in the study of the fragment of arithmetic Normal Open Induction (NOI).NOI is the V3-theory in the language <£ which consists of NZR together with Vx((0(x,O) Λ My > O(0(x,y) -0(x,y + 1)) -Vy > O0(x,y)) for every quantifier-free <£-formula 0(x, y) (x denotes an «-tuple (x ϊ9 ... ,x n )).In [7] Shepherdson gave the following useful characterization of models of NOI:Let M be a normal discretely ordered ring.Then M is a model of NOI if and only if for every element a of the real closure of the fraction field of M there is an element amMsuch that |a -a\ < 1.From this several corollaries are deduced.Let us mention some of them.Let Mbe a model of NOI.Then every quantifier-free definable set in Mis a finite
Abstract We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a …
Abstract We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣ f(x,y) ∣ < 1 for every (x,y) Є C , where C is the curve defined on the real closure of M by C : x 2 + y 2 = a and a > 0 is a nonstandard element of M .
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.
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.
Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties …
Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φ M to φ R and vice versa. Then, we apply these transfer results to give a new proof of a result of M . Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.
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:
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.
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.
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.
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).
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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.
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|.
Abstract We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a …
Abstract We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣ f(x,y) ∣ < 1 for every (x,y) Є C , where C is the curve defined on the real closure of M by C : x 2 + y 2 = a and a > 0 is a nonstandard element of M .
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.
The present work may perhaps be seen as a point of convergence of two historically distinct sequences of results. One sequence of results started with the work of Tennenbaum [59] …
The present work may perhaps be seen as a point of convergence of two historically distinct sequences of results. One sequence of results started with the work of Tennenbaum [59] who showed that there could be no nonstandard recursive model of the system PA of first order Peano arithmetic. Shepherdson [65] on the other hand showed that the system of arithmetic with open induction was sufficiently weak to allow the construction of nonstandard recursive models. Between these two results there remained for many years a large gap occasioned by a general lack of interest in weak systems of arithmetic. However Dana Scott observed that the addition alone of a nonstandard model of PA could not be recursive, while more recently McAloon [82] improved these results by showing that even for the weaker system of arithmetic with only bounded induction, neither the addition nor the multiplication of a nonstandard model could be recursive. Another sequence of results starts with the work of Lessan [78], and independently Jensen and Ehrenfeucht [76], who showed that the structures which may be obtained as the reducts to addition of countable nonstandard models of PA are exactly the countable recursively saturated models of Presburger arithmetic. More recently, Cegielski, McAloon and the author [81] showed that the above result holds true if PA is replaced by the much weaker system of bounded induction. However in both the case of the Tennenbaum phenomenon and in that of the recursive saturation of addition the problem remained open as to how strong a system was really necessary to generate the required phenomenon. All that was clear a priori was that open induction was too weak to produce either result.
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in …
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
Let N be an o-minimal expansion of a real closed field. We develop cohomology theory for the category of N-definable manifolds and N-definable maps, and use this to solve the …
Let N be an o-minimal expansion of a real closed field. We develop cohomology theory for the category of N-definable manifolds and N-definable maps, and use this to solve the Peterzil-Steinhorn problem on the existence of torsion points on N-definably compact N-definable abelian groups. We compute the cohomology rings of N-definably compact N-definable groups, and we prove an o-minimal analog of the Poincare duality theorem, the Alexander dualti theorem, the Lefschetz duality theorem and the Lefschetz fixed point theorem.
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.
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.
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 .
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
It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher …
It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible.