A note on well-generated Boolean algebras in models satisfying Martin's axiom

摘要

A Boolean algebra B is well generated, if it has a well-founded sublattice L such that L generates B. Let B be a superatomic Boolean algebra. The rank of B (rk(B)) is defined to be the Cantor Bendixon rank of the Stone space X of B. For every i ≤ rk(B) let λ_i(B) be the number of isolated points in the i's Cantor Bendixon derivative of X. The cardinal sequence of B is defined as λ(B) :+ <λ_i(B):i ≤ rk(X)>. If aB-{0}, then the rank of a (rk(a)) is defined as the rank of the Boolean algebra Ba{bB:ba}. An element a ∈ B - {0} is a generalized atom (a ∈ At()B), if the last cardinal in the cardinal sequence of Ba is 1. Let a, b ∈ At(B). We denote aBb, if rk(a)=rk(b)=rk(a·b). A subset H is contained in At(b) is a complete set of representatives (CSR) for B, if for every there is a unique bH such that bBa. Any CSR for B generates B. We say that B is hereditarily decreasingly canonically well generated, if for every subalgebra C of B and every CSR H for C there is a CSR M for C such that: (1) for every aH and bM: if bCa then b ≤ a; (2) the sublattice of C generated by M is well founded. Theorem. Assume (MA + x_1 < 2~(x_0)). Let ε be a countable ordinal, κ < 2~(x_0) and n < ω. If B is a superatomic Boolean algebra such that λ(B) = <κ, x_1, n> or λ(B) = , then B is hereditarily decreasingly canonically well generated.