On well-generated Boolean algebras

A Boolean algebra B that has a well-founded sublattice L which generates B is called a well-generated (WG) Boolean algebra. If in addition, L is generated by a complete set of representatives for B (see Definition 1.1), then B is said to be canonically well-generated (CWG). Every WG Boolean algebra is superatomic. We construct two basic examples of superatomic non well-generated Boolean algebras. Their cardinal sequences are 〈א₁,א₀,א₁,1〉 and 〈א₀,א₀,2^א0,1〉. Assuming MA∧(א₁<2^א0), we show that every algebra with one of the cardinal sequences 〈א₀:i<α〉̂〈λ,א ₁,1〉, α<א₁,λ<2^א0, or 〈א₀,2^א0,א₁,1〉 is CWG. Assuming CH, or alternatively assuming MA∧(2^א0=א₂), we determine which cardinal sequences admit only WG Boolean algebras. We find a necessary and sufficient condition for the canonical well-generatedness of algebras whose cardinal sequence has the form 〈א₀:i<α〉̂〈λ,1〉, α<א₁. We conclude that if such an algebra is CWG, then all of its quotients are CWG. We show that the above is not true for general Boolean algebras. We also conclude that if the cardinality of such an algebra is less than the cardinal b defined below, then it is CWG. The cardinal b is the least cardinality of an unbounded subset of {f|f:ω→ω}. We investigate questions concerning embeddability, quotients and subalgebras of WG and CWG Boolean algebras, and construct various counter-examples.