More on Cardinal Invariants of Boolean Algebras

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(B0 × B1) = max{irr(B0), irr(B1)}. We prove consistency of the statement “there is a Boolean algebra B such that irr(B) < s(B ~ B)” and we force a superatomic Boolean algebra B∗ such that s(B∗) = inc(B∗) = κ, irr(B∗) = Id(B∗) = κ and Sub(B∗) = 2 + . Next we force a superatomic algebra B0 such that irr(B0) < inc(B0) and a superatomic algebra B1 such that t(B1) > Aut(B1). Finally we show that consistently there is a Boolean algebra B of size λ such that there is no free sequence in B of length λ, there is an ultrafilter of tightness λ (so t(B) = λ) and λ / ∈ DepthHs(B). ∗ Partially supported by “Basic Research Foundation” of the Israel Academy of Sciences and Humanities. Publication 599.