# Completion of the countable Cantor algebra?

My question arises from the article "Cantor algebra" in Wikipedia.

I quote:

"The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Boolean algebra on a countable number of generators. Up to isomorphism, this is the only nontrivial Boolean algebra that is both countable and atomless.

The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets (Balcar & Jech 2006). It is isomorphic to the completion of the countable Cantor algebra..."

Wikipedia does not precise what is the completion in question. So, what is the completion of the countable Cantor algebra? Is it the Boolean algebra of regular open sets of the Cantor set?

• I believe the completion that you're looking for is Dedekind-MacNeille completion. – Eran Nov 5 '17 at 14:46

A completion of a Boolean algebra $B$ is a complete Boolean algebra $C$ together with a complete homomorphism $f:B\to C$ such that for any other complete homomorphism $g:B\to D$ to a complete Boolean algebra $D$, there is a unique complete homomorphism $h:C\to D$ such that $g=hf$. Here a "complete homomorphism" is a Boolean homomorphism that preserves all joins and meets that exist in the domain. By the usual abstract nonsense, the completion of $B$ is unique up to unique isomorphism commuting with the maps from $B$.
In concrete terms, a completion of any Boolean algebra $B$ can be constructed as the regular open algebra of the Stone space of $B$. So, as you say, the completion of the Cantor algebra is the regular open algebra of the Cantor set. The regular open algebra of a Stone space (or indeed of any compact Hausdorff space) is also isomorphic to the algebra of Borel sets module meager sets, thus giving the description in Wikipedia. The completion can also be described as the Dedekind-MacNeille completion of $B$ as a poset, as Henno Brandsma mentioned.