[FOM] complete atomless boolean algebras

[James Hirschorn]

On Thursday 17 August 2006 19:24, Robert Black wrote:
> I have two questions about complete atomless boolean algebras to
> which I find I can't (or can't easily) get answers from the
> Koppelberg Handbook, but I expect there are members of this list who
> can answer them instantly:
>
> 1) An infinite complete boolean algebra must have a cardinality k
> such that k=k^aleph_0. Is this the only cardinality restriction on
> complete *atomless* Boolean algebras? In particular, can the 
> cardinality of a complete atomless Boolean algebra be inaccessible?
>
> 2) A Boolean algebra B is homogeneous (I think this is the standard
> word) iff for every nonzero p in B, the algebra induced on the x in B
> less than or equal to p by the partial order inherited from B is
> isomorphic to the whole of B. Trivially a homogeneous Boolean algebra
> with cardinality greater than 2 is atomless. Now consider *complete*
> homogeneous Boolean algebras. (There are such things, since unless
> I'm making an embarrassing mistake the regular open algebra over R is
> one.

Yes. R can be taken as the supremum of a countable family of pairwise disjoint 
open intervals, and so can any regular open subset of R. There is a natural 
isomorphism resulting from this observation.

> ) Same question again: what cardinalities can complete 
> homogeneous Boolean algebras have, and in particular can their
> cardinality be inaccessible?

Sorry, I can't help you with this. I frequently use complete homogeneous 
Boolean algebras, but I don't normally concern myself with their cardinality. 


James Hirschorn
>
> Robert
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom