On essentially low, canonically well-generated boolean algebras

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 ⌈a =^def (b ∈ B: B ≤ a). The rank of 0^B is defined to be -1. An element a ∈ B - (0) is a generalized atom (a ∈ At(B)), if the last nonzero cardinal in the cardinal sequence of B ⌈a is 1. Let a, b ∈ At(B). We denote a ∼ b, if rk(a) = rk(b) = rk(a. b). A subset H ⊆ At(B) is a complete set of representatives (CSR) for B, if for every a ∈ 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 (N₀: I < α)λ, 1), a < N₁. 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.