# Sh:530

• Cummings, J., & Shelah, S. (1995). A model in which every Boolean algebra has many subalgebras. J. Symbolic Logic, 60(3), 992–1004.
• 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. 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.
• published version (14p)
Bib entry
@article{Sh:530,
author = {Cummings, James and Shelah, Saharon},
title = {{A model in which every Boolean algebra has many subalgebras}},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {60},
number = {3},
year = {1995},
pages = {992--1004},
issn = {0022-4812},
doi = {10.2307/2275769},
mrclass = {03E35 (03G05)},
mrnumber = {1349006},
mrreviewer = {Piotr Koszmider},
doi = {10.2307/2275769},
note = {\href{https://arxiv.org/abs/math/9509227}{arXiv: math/9509227}},
arxiv_number = {math/9509227}
}