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

Theorem.Assume(MA+ℵ1<2ℵ0). Letεbe a countable ordinal, κ<2ℵ0andn<ω. If B is a superatomic Boolean algebra such thatλ⇒(B)=〈ℵ0:i<ε〉^〈κ,ℵ1,n〉orλ⇒(B)=〈ℵ0,2ℵ0,ℵ1,n〉, then B is hereditarily decreasingly canonically well generated.