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.
In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer …
In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer to as "definable groups"). It is known (\cite{Pi88}) that any group definable in an o-minimal expansion of the real field is a Lie group, and in \cite{COP} a complete characterization of when a Lie group has a "definable group" which is \emph{Lie isomorphic} to it was given. We continue the analysis by explaining when a Lie homomorphism between definable groups is a definable isomorphism. Among other things, we prove that in any o-minimal expansion $\mathcal R$ of the real field we can add a function symbol for any Lie isomorphism between definable groups to the language of $\mathcal R$ preserving o-minimality, and that any definable group $G$ can be endowed with an analytic manifold structure definable in $\mathcal R_{\text{Pfaff}}$ that makes it an analytic group.
Real closed fields, and structures associated with them, are interesting from the point of view of both model theory and computability. In this paper, we give results on the complexity …
Real closed fields, and structures associated with them, are interesting from the point of view of both model theory and computability. In this paper, we give results on the complexity of value group sections and residue field sections. It is not difficult to show that for any countable real closed field R, there is a value group section that is Δ 2 0 ( R ) . This result is sharp in the sense that there is a computable real closed field for which every value group section codes the halting set. For a real closed field R, there is a residue field section that is Π 2 0 ( R ) . This result is sharp in the sense that there is a computable real closed field R with no Σ 2 0 value field section. We are also interested in integer parts. Mourgues and Ressayre showed, by a rather complicated construction, that every real closed field has an integer part. The construction becomes canonical once we fix the real closed field R, a residue field section k, and a well ordering of R. The construction involves mapping the elements of R to generalized series, called developments, with terms corresponding to elements of the natural value group and coefficients in k. The complexity of the construction is clearly related to the lengths of the developments. We conjecture that for a type ω well ordering on R, the lengths of the developments are less than ω ω ω . We give an example showing that there is no smaller ordinal bound.
A classical theorem of Steinitz [12, p. 125] states that the characteristic of an algebraically closed field, together with its absolute degree of transcendency, uniquely determine the field (up to …
A classical theorem of Steinitz [12, p. 125] states that the characteristic of an algebraically closed field, together with its absolute degree of transcendency, uniquely determine the field (up to isomorphism). It is easily seen that the word real-closed cannot be substituted for the words algebraically closed in this theorem. It is therefore natural to inquire what invariants other than the absolute transcendence degree are needed in order to characterize a real-closed field.' For non-denumerable fields, the question is equivalently stated as follows: what invariants in addition to the cardinal number are needed in order to characterize a real-closed field? Now, it is well-known that any two isomorphic realclosed fields are similarly ordered (i.e., as ordered sets). Here we establish the converse implication2 for a particular class of non-denumerable,3 non-archimedean, real-closed fields. Section 2 of our paper is devoted to the proof of this theorem (Theorem 2.1). The class of ordered fields to which our isomorphism theorem applies is quite restricted. (In fact, in order that it not be vacuous, we must assume either the continuum hypothesis, or some one of its generalizations to higher cardinals.4) Nevertheless, we are able to find an application to a class of fields that is not insignificant, namely, those that appear as residue class fields of maximal ideals in rings of continuous functions (on completely regular topological spaces). This discussion is the content of Section 3, and leads to the theorem that all nonarchimedean residue class fields (the so-called hyper-real fields) of power R, are isomorphic (Theorem 3.5). As a rather interesting corollary to this theorem, we find (using the continuum hypothesis) that all the non-real residue class fields of maximal ideals of a countable complete direct sum of real fields are isomorphic (Corollary 3.9). Section 4 continues the discussion of non-archimedean residue class fields. The development here leads to the construction of various such fields that arise from the same ring, but have different cardinal numbers (Theorems 4.4 ff. and 4.8 ff.). (A fortiori, not all such fields that arise from the same ring are isomorphic.) This section is almost entirely set-theoretic in character, and some of the results obtained here have some set-theoretic interest in themselves (Lemmas 4.1 and 4.7). (No use is made of the continuum hypothesis in this section.) Finally, in Section 5, we pose some unsolved problems.
A. Grothendieck introduced the notion of “tame geometry” in [8], more precisely in a chapter entitled “Denunciation of so-called general topology, and heuristic reflexions towards a so-called tame topology”. He …
A. Grothendieck introduced the notion of “tame geometry” in [8], more precisely in a chapter entitled “Denunciation of so-called general topology, and heuristic reflexions towards a so-called tame topology”. He says there that general topology has been “developed by analysts in order to meet the needs of analysis”, and “not for the study of topological properties of the various geometrical shapes”. Consequently, according to him, when one tries to work in the technical context of topological spaces, “one is confronted at each step with spurious difficulties related to wild phenomena”.
Using a modification of Wilkie''s recent proof of o-minimality for Pfaffian functions, we gave an invariant characterization of o-minimal expansions of IR. We apply this to construct the Pfaffian closure …
Using a modification of Wilkie''s recent proof of o-minimality for Pfaffian functions, we gave an invariant characterization of o-minimal expansions of IR. We apply this to construct the Pfaffian closure of an arbitrary o-minimal expansion of IR.
Here we study algebraic function fields K, give necessary and sufficient condition for the ideal class group $H(K)$ of any real quadratic function field $K$ to have a cyclic subgroup …
Here we study algebraic function fields K, give necessary and sufficient condition for the ideal class group $H(K)$ of any real quadratic function field $K$ to have a cyclic subgroup of order $n$, and obtain eight series of such fields $K$, with four of them NOT ERD-type or GERD-type.
Here we study algebraic function fields K, give necessary and sufficient condition for the ideal class group $H(K)$ of any real quadratic function field $K$ to have a cyclic subgroup …
Here we study algebraic function fields K, give necessary and sufficient condition for the ideal class group $H(K)$ of any real quadratic function field $K$ to have a cyclic subgroup of order $n$, and obtain eight series of such fields $K$, with four of them NOT ERD-type or GERD-type.
We establish basic properties of differential topology for defianable manifolds in an o-minimal expansion M of R = (R,+;., <), where 0 < < and is a definable group.
We establish basic properties of differential topology for defianable manifolds in an o-minimal expansion M of R = (R,+;., <), where 0 < < and is a definable group.
We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order …
We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb{Q}(\zeta_n)$ to one over $E$.
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and …
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.
Abstract In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory …
Abstract In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G -compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a , the Lascar group of $\operatorname {stp}(ac)$ is abelian, and the underlying theory is G -compact, then the Lascar groups of $\operatorname {stp}(ac)$ and of $\operatorname {stp}(a)$ are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of $\operatorname {stp}(ac)$ is abelian, nor the assumption of c being finite can be removed.
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:
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a …
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.
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 .
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg …
This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.
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.
It is the aim of this work to study product structures on four dimensional solvable Lie algebras.We determine all possible paracomplex structures and consider the case when one of the …
It is the aim of this work to study product structures on four dimensional solvable Lie algebras.We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal.These results are applied to the case of Manin triples and complex product structures.We also analyze the three dimensional subalgebras.
We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable …
We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable groups. We also present various results on denable modules and actions, denable extensions and
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment …
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.
Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable.
Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C 1 coefficients are definable.
In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.