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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  and
Saharon Shelah
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
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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 2∣A∣ = 2∣B∣. This implies in particular that B has 2∣B∣ 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.

Keywords

Boolean algebrasfree subsetsRadin forcing
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 3 , September 1995 , pp. 992 - 1004
Copyright
Copyright © Association for Symbolic Logic 1995

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References

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