Chains of well-generated Boolean algebras whose union is not well-generated

A Boolean algebra B that has a well-founded sublattice which generates B is called a well-generated Boolean algebra. Every well-generated Boolean algebra is superatomic. However, there are superatomic algebras which are not well-generated. We consider two types of increasing chains of Boolean algebras, canonical chains and rank preserving chains, and show that the class of well-generated Boolean algebras is not closed under union of such chains, even when these chains are taken to be countable. A Boolean algebra is superatomic iff its Stone space is scattered. If B is superatomic and a ∈ B, then the rank of a is the Cantor Bendixon rank of the Stone space of {b} b ≤ a}. A chain {B_α α << δ} is a canonical chain if for every α < β <, δ, B_α is the subagebra of B_β generated by all members of B_β whose rank is < α. For a superatomic algebra B, I(B) denotes the ideal consisting of all members of B whose rank is less than the rank of B. A chain {B_α| α < δ} is a rank preserving chain if for every α < β < δ and α ∈ I(B _α), the rank and mutiplicity of a in B_α are equal to the rank and mutiplicity of a in B_β.