Re: Atomless Boolean Algebra

From: Michael J Hardy (mjhardy_at_mit.edu)
Date: 07/23/04

  • Next message: Yves de Cornulier: "Re: embeddability of a finite metric space in a Riemannian manifold" 
    Date: 23 Jul 2004 20:40:41 GMT

    William Elliot (marsh@agora.rdrop.com) wrote:
                                                                                    
    > Some questions arise for which I request your suggestions.
    > Is this Boolean algebra incomplete?
    > Is there an example showing its incompleteness?
    > Is there a way to complete it?

    Someone else answered much of this. I would add that
    every Boolean algebra has a Dedekind completion.

    Roman Sikorski has written a book on Boolean algebras
    that gives an account of this.

    Paul Halmos wrote _Lectures_on_Boolean_Algeras_, which
    has two flaws:

    (1) He says "non-atomic" (or maybe "nonatomic"?) instead
        of "atomless". The word is confusing because it sounds
        as if it means "not atomic". I think he got this from
        the language of ergodic theory.

    (2) In his section on completions, Lecture 21, he seems to
        be under the impression that a Boolean algebra has
        various non-isomorphic "completions", one of which is
        "minimal". But even though he's got it wrong, one can
        reconstruct the truth from what he wrote.

    Halmos' book is generally very readable, and despite those
    two flaws, a good way to learn this topic. -- Mike Hardy

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