On well-generated Boolean algebras Original Research Article

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 View the MathML source and View the MathML source. Assuming View the MathML source, we show that every algebra with one of the cardinal sequences View the MathML source, View the MathML source, or View the MathML source is CWG. Assuming CH, or alternatively assuming View the MathML source, 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 View the MathML source, View the MathML source. 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 View the MathML source defined below, then it is CWG. The cardinal View the MathML source 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.