Superatomic Boolean algebras constructed from strongly unbounded functions

✍ Scribed by Juan Carlos Martínez; Lajos Soukup

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 186 KB
• Volume: 57
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.201010014

✦ Synopsis

Using Koszmider's strongly unbounded functions, we show the following consistency result:

Suppose that κ, λ are infinite cardinals such that κ + + + ≤ λ, κ < κ = κ and 2 κ = κ + , and η is an ordinal with κ + ≤ η < κ + + and cf(η) = κ + . Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra B such that ht(B) = η + 1, wdα (B) = κ for every α < η and wdη (B) = λ (i.e., there is a locally compact scattered space with cardinal sequence κ η λ ).

Especially, ω ω 1 ω3 and ω1 ω 2 ω4 can be cardinal sequences of superatomic Boolean algebras.