On well-generated Boolean algebras

✍ Scribed by Robert Bonnet; Matatyahu Rubin

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 390 KB
• Volume: 105
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(99)00034-2

✦ Synopsis

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 Deÿnition 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 ℵ 0 ; 1 .

Assuming MA ∧ (ℵ1¡2 ℵ 0 ), we show that every algebra with one of the cardinal sequences ℵ0: i¡ ˆ ; ℵ1; 1 , ¡ℵ1; ¡2 ℵ 0 , or ℵ0; 2 ℵ 0 ; ℵ1; 1 is CWG.

Assuming CH, or alternatively assuming MA ∧ (2 ℵ 0 = ℵ2), we determine which cardinal sequences admit only WG Boolean algebras.

We ÿnd a necessary and su cient condition for the canonical well-generatedness of algebras whose cardinal sequence has the form ℵ0: i¡ ˆ ; 1 , ¡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 deÿned below, then it is CWG. The cardinal b is the least cardinality of an unbounded subset of {f | f : ! → !}.

We investigate questions concerning embeddability, quotients and subalgebras of WG and CWG Boolean algebras, and construct various counter-examples.