On essentially low, canonically well-generated Boolean algebras

Journal of Symbolic Logic 67 (1):369-396 (2002)
Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)), is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B - {0}, then the rank of a in B (rk(a)), is defined to be the rank of the Boolean algebra $B b \upharpoonright a \overset{\mathrm{def}}{=} \{b \in B: b \leq a\}$ . The rank of 0 B is defined to be -1. An element a ∈ B - {0} is a generalized atom $(a \in \widehat{At}(B))$ , if the last nonzero cardinal in the cardinal sequence of B $\upharpoonright$ a is 1. Let a,b $\in\widehat{At}$ (B). We denote a ∼ b, if rk(a) = rk(b) = rk(a · b). A subset H $\subseteq \widehat{At}$ (B) is a complete set of representatives (CSR) for B, if for every a $\in \widehat{At}$ (B) there is a unique h ∈ H such that h ∼ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B. THEOREM 1. Let B be a Boolean algebra with cardinal sequence $\langle\aleph_0: i . If B is CWG, then every subalgebra of B is CWG. A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1. Theorem 1 follows from Theorem 2.9, which is the main result of this work. For an ESL BA B we define a set F B of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent. (1) Every subalgebra of B is CWG; and (2) F B is bounded. THEOREM 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated
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