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The completeness of the isomorphism relation for countable Boolean algebras

Authors: Riccardo Camerlo and Su Gao
Journal: Trans. Amer. Math. Soc. 353 (2001), 491-518
MSC (2000): Primary 03E15, 06E15
Published electronically: September 21, 2000
MathSciNet review: 1804507
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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 first-order 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.

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Additional Information

Riccardo Camerlo
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Su Gao
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Keywords: Borel reducibility, polish group actions, definable equivalence relations, separable Boolean spaces
Received by editor(s): March 11, 1999
Published electronically: September 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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