A model in which every infinite Boolean algebra has many subalgebras
逻辑
2009-09-25 v1
摘要
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.
引用
@article{arxiv.math/9509227,
title = {A model in which every infinite Boolean algebra has many subalgebras},
author = {James Cummings and Saharon Shelah},
journal= {arXiv preprint arXiv:math/9509227},
year = {2009}
}