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

Abstract

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 a ∈ B - { 0 } , then the rank of a ( rk ( a ) ) is defined as the rank of the Boolean algebra B ↾ a ≔ { b ∈ B : b ⩽ a } . An element a ∈ B - { 0 } is a generalized atom ( a ∈ At ^ ( B ) ), if the last cardinal in the cardinal sequence of B ↾ a is 1. Let a , b ∈ At ^ ( B ) . We denote a ∼ B 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 b ∈ H such that b ∼ B a . 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 a ∈ H and b ∈ M : if b ∼ C a then b ⩽ a ; (2) the sublattice of C generated by M is well founded. Theorem. Assume ( MA + ℵ 1 < 2 ℵ 0 ) . Let ε be a countable ordinal, κ < 2 ℵ 0 and n < ω . If B is a superatomic Boolean algebra such that λ ⇒ ( B ) = 〈 ℵ 0 : i < ε 〉 ^ 〈 κ , ℵ 1 , n 〉 or λ ⇒ ( B ) = 〈 ℵ 0 , 2 ℵ 0 , ℵ 1 , n 〉 , then B is hereditarily decreasingly canonically well generated.