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Borel complexity of isomorphism between quotient Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Su Gao  and
Michael Ray Oliver
Su Gao
Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203–1430, USA, E-mail: [email protected], URL: http://www.math.yorku.ca/~moliver
Michael Ray Oliver
Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203–1430, USA, E-mail: [email protected], URL: http://www.math.yorku.ca/~moliver
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Extract

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:

(i) “Borel ideal” may be improved to “analytic P-ideal”

(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.

See [Oli04].

In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1328 - 1340
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[AdKech00]Adams, Scot and Kechris, Alexander S., Linear algebraic groups and countable Borel equivalence relations, Journal of the American Mathematical Society, vol. 13 (2000), no. 4, pp. 909–943.CrossRefGoogle Scholar
[Far04]Farah, Ilijas, How many Boolean algebras are there?, Illinois Journal of Mathematics, vol. 46 (2002), no. 4, pp. 999–1033.CrossRefGoogle Scholar
[Mosch80]Moschovakis, Yiannis, Descriptive set theory, North-Holland, 1980.Google Scholar
[Oli04]Oliver, Michael Ray, Continuum-many Boolean algebras of the form , I Borel, this Journal, vol. 69 (2004), pp. 799–816.Google Scholar