On tightness and depth in superatomic Boolean algebras

Abstract

We introduce a large cardinal property which is consistent with L and show that for every superatomic Boolean algebra B and every cardinal λ with the large cardinal property, if tightness+(B) ≥ λ+, then depth(B) ≥ λ. This improves a theorem of Dow and Monk. In [DM, Theorem C], Dow and Monk have shown that if λ is a Ramsey cardinal (see [J, p.328]), then every superatomic Boolean algebra with tightness at least λ has depth at least λ. Recall that a Boolean algebra B is superatomic iff every homomorphic image of B is atomic. The depth of B is the supremum of all cardinals λ such that there is a sequence (bα : α < λ) in B with bβ < bα for all α < β < λ (a well-ordered chain of length λ). Then depth of B is the first cardinal λ such that there is no well-ordered chain of length λ in B. The tightness of B is the supremum of all cardinals λ such that B has a free sequence of length λ, where a sequence (bα : α < λ) is called free provided that if Γ and ∆ are finite subsets of λ such that α < β for all α ∈ Γ and β ∈ ∆, then ⋂