Results 1  10
of
47
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
The classification of hypersmooth Borel equivalence relations
 J. Amer. Math. Soc
, 1997
"... This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can ..."
Abstract

Cited by 30 (4 self)
 Add to MetaCart
This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel automorphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bireducibility, and Borel isomorphism. An equivalence relation E on X is Borel reducible to an equivalence relation F on Y if there is a Borel map f: X → Y with xEy ⇔ f(x)Ff(y). We write then E ≤ F. If E ≤ Fand F ≤ E we say that E,F are Borel bireducible, in symbols E ≈ ∗ F.When E ≈ ∗ Fthe quotient spaces X/E, Y/F have the same “effective ” or “definable ” cardinality. We say that E,F are Borel isomorphic if there exists a Borel bijection f: X → Y with xEy ⇔ f(x)Ff(y). Below we denote by E0,Et the equivalence relations on the Cantor space 2 N given by: xE0y ⇔
The classification problem for torsionfree abelian groups of finite rank
 J. Amer. Math. Soc
, 2001
"... In 1937, Baer [5] introduced the notion of the type of an element in a torsionfree abelian group and showed that this notion provided a complete invariant for the classification problem for torsionfree abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] ..."
Abstract

Cited by 21 (8 self)
 Add to MetaCart
In 1937, Baer [5] introduced the notion of the type of an element in a torsionfree abelian group and showed that this notion provided a complete invariant for the classification problem for torsionfree abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory
On the Complexity of the Isomorphism Relation for Finitely Generated Groups
, 1998
"... Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two genera ..."
Abstract

Cited by 20 (11 self)
 Add to MetaCart
Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators. The proofs are grouptheoretic, and we refer to descriptive set theory only for the relevant definitions and for motivation for the results. Introduction Given a class K of structures for a fixed first order language L, one may ask what kinds of complete invariants can be used to classify the elements of K up to isomorphism. For those classes consisting of the countable models of some L ! 1 ;! sentence, Friedman and Stanley [FS] proposed to use the methods of descriptive set theory to study their possible invariants and defined the notion of Borel reducibility between such classes of structures. In [HK], Hjorth and Kechris continued this study and situated it within the general the...
Superrigidity and countable Borel equivalence relations
 Annals Pure Appl. Logic
"... Introduction. These notes are based upon a daylong lecture workshop presented by Simon ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
Introduction. These notes are based upon a daylong lecture workshop presented by Simon
The classification problem for plocal torsionfree abelian groups of finite rank
, 2002
"... Let n ≥ 3. We prove that if p != q are distinct primes, then the classification problems for plocal and qlocal torsionfree abelian groups of rank n are incomparable with respect to Borel reducibility. ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Let n ≥ 3. We prove that if p != q are distinct primes, then the classification problems for plocal and qlocal torsionfree abelian groups of rank n are incomparable with respect to Borel reducibility.
The completeness of the isomorphism relation for countable Boolean algebras
 Trans. Amer. Math. Soc
"... Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is o ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete firstorder theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.
Polish group actions: dichotomies and generalized elementary embeddings
 J. Amer. Math. Soc
, 1997
"... The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same. ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Linear algebraic groups and countable Borel equivalence relations
 J. AMER. MATH. SOC
, 1999
"... This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these obje ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c: X → I such that xEy ⇔ c(x) =c(y). For this to be of any interest both I and c must be explicit or definable and as simple and concrete as possible. The theory of Borel equivalence relations studies the settheoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems. In organizing this study, the following concept of reducibility is fundamental. Let E,F be equivalence relations on standard Borel spaces X, Y, resp. We say that E is Borel reducible to F,insymbols, E ≤B F,