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A model in which every Boolean algebra has many subalgebras

Published online by Cambridge University Press:  12 March 2014

James Cummings
Affiliation:
Mathematics Institute, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel, E-mail: cummings@math.huji.ac.il
Saharon Shelah
Affiliation:
Mathematics Institute, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel, E-mail: shelah@math.huji.ac.il

Abstract

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2A = 2B. This implies in particular that B has 2B subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra.

The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a “black box” at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1995

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References

[1]Baumgartner, J. and Taylor, A., Saturation properties of ideals in generic extensions, I, Transactions of the American Mathematical Society, vol. 270 (1982), pp. 557574.CrossRefGoogle Scholar
[2]Foreman, M. and Woodin, H., GCH can fail everywhere, Annals of Mathematics, vol. 133 (1991), pp. 135.CrossRefGoogle Scholar
[3]Jech, T., Set theory, Academic Press, 1978.Google Scholar
[4]Laver, R., Making the supercompactness of K indestructible under κ-directed-closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.CrossRefGoogle Scholar
[5]Magidor, M., On the singular cardinals problem I, Israel Journal of Mathematics, vol. 28 (1977), pp. 131.CrossRefGoogle Scholar
[6]Monk, J. D., Cardinal functions on Boolean algebras, Birkhaüser, 1990.CrossRefGoogle Scholar
[7]Monk, J. D. and Bonnet, Robert (editors), Handbook of Boolean algebras, North-Holland, 1989.Google Scholar
[8]Radin, L. B., Adding closed cofinal sequences to large cardinals, Annals of Mathematical Logic, vol. 22 (1982), pp. 243261.CrossRefGoogle Scholar
[9]Shelah, S., Cardinal arithmetic, Oxford University Press, to appear.Google Scholar
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