More on cardinal invariants of Boolean algebras

Andrzej Roslanowski, Saharon Shelah

Published 1998-08-12Version 1

We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove consistency of the statement ``there is a Boolean algebra B such that irr(B)<s(B otimes B)'' and we force a superatomic Boolean algebra B_* such that s(B_*)=inc(B_*)=kappa, irr(B_*)=Id(B_*)=kappa^+ and Sub(B_*)=2^(kappa^+). Next we force a superatomic algebra B_0 such that irr(B_0)<inc(B_0) and a superatomic algebra B_1 such that t(B_1)>Aut(B_1). Finally we show that consistently there is a Boolean algebra B of size lambda such that there is no free sequence in B of length lambda, there is an ultrafilter of tightness lambda (so t(B)=lambda) and lambda notin Depth_(Hs)(B).