In this paper, we characterize the possible cofinalities of the least $\lambda $-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta $, that carries a $\lambda $-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $\delta $, that the least $\lambda $-strongly compact cardinal has cofinality $\delta $. On the other hand, provably the cofinality of the least $\lambda $-strongly compact cardinal always carries a $\lambda $-complete uniform ultrafilter.

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.