Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-00-02659-3
Published electronically: September 21, 2000
MathSciNet review: 1804507
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 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.


References [Enhancements On Off] (What's this?)

  • [An58] R. D. Anderson, The algebraic simplicity of certain groups of homeomorphisms, American Journal of Mathematics 80 (1958), 955-963. MR 20:4607
  • [Ba75] J. Barwise, Admissible Sets and Structures: An Approach to Definability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975. MR 54:12519
  • [BK96] H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Notes Series 232, Cambridge University Press, 1996. MR 98d:54068
  • [Bl98] B. Blackadar, K-theory for Operator Algebras, Second Edition, Cambridge University Press, 1998. MR 99g:46104
  • [Br72] O. Bratteli, Inductive limits of finite dimensional $C^*$-algebras, Transactions of the American Mathematical Society 171 (1972), 195-234. MR 47:844
  • [FS89] H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, The Journal of Symbolic Logic 54 (1989), 894-914. MR 91f:03062
  • [Ga98] S. Gao, The isomorphism relation between countable models and definable equivalence relations, PhD dissertation, UCLA, 1998.
  • [Go97] S. S. Goncharov, Countable Boolean Algebras and Decidability, Consultants Bureau, 1997. MR 98h:03044b
  • [Ha76] W. Hanf, Representing real numbers in denumerable Boolean algebras, Fundamenta Mathematicae 91 (1976), 167-170. MR 54:7252
  • [Hj98] G. Hjorth, Classification and Orbit Equivalence Relations, Mathematical Surveys and Monographs, 75, Amer. Math. Soc., Providence, RI, 2000. CMP 2000:05
  • [HK95] G. Hjorth and A. S. Kechris, Analytic equivalence relations and Ulm type classifications, The Journal of Symbolic Logic 60 (1995), 1273-1300. MR 96m:54068
  • [Ho93] W. Hodges, Model Theory, Cambridge University Press, 1993. MR 94e:03002
  • [Iv91] P. Iverson, The number of countable isomorphism types of the theory of Boolean algebras, Colloquium Mathematicum 62 (2) (1991), 181-187. MR 93a:03038
  • [Kec95] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995. MR 96e:03057
  • [Kec98] A. S. Kechris, The descriptive classification of some classes of $C^*$-algebras, Proceedings of the Sixth Asian Logic Conference (1998), 121-149.
  • [Ket78] J. Ketonen, The structure of countable Boolean algebras, Annals of Mathematics 108 (1978), 41-89. MR 58:10647
  • [Ko89] S. Koppelberg, Handbook of Boolean Algebras, vol. 1 (J.D. Monk ed.), North-Holland, 1989. MR 90k:06002
  • [Mu90] G. J. Murphy, $C^*$-algebras and Operator Theory, Academic Press, 1990. MR 91m:46084
  • [Na72] M. A. Naimark, Normed Algebras, Wolters-Noordhoff, 1972. MR 55:11042
  • [Pa95] G. Panti, La logica infinito-valente di \Lukasiewicz, PhD dissertation, Università degli studi di Siena, 1995.
  • [Pi89] R. S. Pierce, Countable Boolean algebras, in Handbook of Boolean Algebras (J. D. Monk and R. Bonnet eds.), Elsevier Science Publishers, 1989, 775-876. CMP 21:10
  • [vM89] J. van Mill, Infinite Dimensional Topology. Prerequisites and Introduction, Elsevier Science Publishers, 1989. MR 90a:57025

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E15, 06E15

Retrieve articles in all journals with MSC (2000): 03E15, 06E15


Additional Information

Riccardo Camerlo
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email: camerlo@its.caltech.edu

Su Gao
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Email: sugao@its.caltech.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02659-3
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

American Mathematical Society