A MODEL IN WHICH EVERY BOOLEAN-ALGEBRA HAS MANY SUBALGEBRAS

We show that it is consistent with ZFC (relative to large cardinals) t hat every infinite Boolean algebra B has an irredundant subset A such that 2(\A\) = 2(\B\). This implies iri particular that B has 2(\B\) su balgebras. We also discuss some more general problems about subalgebra s and free subsets of an algebra. The result on the number of subalgeb ras 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 ''bl ack box'' at the end. The proof involves a variation on Foreman and Wo odin's model in which GCH fails everywhere.