The Isomorphism Relation Between Tree-Automatic Structures

An $\omega$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $\omega$-tree-automatic structures. We prove first that the isomorphism relation for $\omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $\omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a $\Sigma_2^1$-set nor a $\Pi_2^1$-set

The almost isomorphism relation for simple regular rings

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K.C . O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective modules over a simple regular ring

Borel reducibility and classification of von Neumann algebras

We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras

Borel Complexity of the Isomorphism Relation for O-minimal Theories

In 1988, Mayer published a strong form of Vaught\u27s Conjecture for o-minimal theories (1). She showed Vaught\u27s Conjecture holds, and characterized the number of countable models of an o-minimal theory T if T has fewer than 2K ° countable models. Friedman and Stanley have shown in (2) that several elementary classes are Borel complete. This work addresses the class of countable models of an o-minimal theory T when T has 2N ° countable models, including conditions for when this class is Borel complete. The main result is as follows. Theorem 1. Let T be an o-minimal theory in a countable language having 2N ° countable models. Either i. For every finite set A, every p(x) E S1 (A) is simple, and isomorphism on the class of countable models of T is ∏03 (and is, in fact, equivalence of countable sets of reals); or ii. For some finite set A, some p(x) E S1 (A) is non-simple, and there is a finite set B D A such that the class of countable models of T over B is Borel complete