Partition Complete Boolean Algebras and Almost Compact Cardinals

✍ Scribed by Peter Jipsen; Henry Rose

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 935 KB
• Volume: 45
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19990450208

✦ Synopsis

Abstract

For an infinite cardinal K a stronger version of K‐distributivity for Boolean algebras, called k‐partition completeness, is defined and investigated (e. g. every K‐Suslin algebra is a K‐partition complete Boolean algebra). It is shown that every k‐partition complete Boolean algebra is K‐weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K‐partition complete Boolean algebra is an updirected union of basic K‐tree algebras. Using K‐partition completeness, the concept of γ‐almost compactness is introduced for γ ≥ K. For strongly inaccessible K we show that K is K‐almost compact iff K is weakly compact, and if K is 2^K^‐almost compact, then K is measurable. Further K is strongly compact iff it is γ‐almost compact for all γ ≥ K.