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

Abstract

A Boolean algebraB that has a well-founded sublattice which generatesB is called awell-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 issuperatomic iff its Stone space is scattered. IfB is superatomic anda∈B, then therank ofa is the Cantor Bendixon rank of the Stone space of{b‖b≤a}. A chain {B _α‖α<δ} is acanonical chain if for every α<β<δ,B _αis the subagebra ofB _βgenerated by all members ofB _βwhose rank is <α. For a superatomic algebraB, I(B) denotes the ideal consisting of all members ofB whose rank is less than the rank ofB. A chain {B _α‖α<δ} is arank preserving chain if for every α<β<δ anda∈I(B_α), the rank and mutiplicity ofa inB _αare equal to the rank and mutiplicity ofa inB _β.