Is there such a thing as the sigma-completion of a Boolean algebra?

Question

Hi all,

Suppose that $\mathcal{B}$ is a Boolean algebra. It there a way to extend $\mathcal{B}$ to a smallest Boolean algebra $\mathcal{B}'$ that contains an isomorphic copy of $\mathcal{B}$ and is countably complete, i.e. every countable subset of $\mathcal{B}'$ has a least upper bound in $\mathcal{B}'$? By "smallest" I mean that the inclusion $i: \mathcal{B} \hookrightarrow \mathcal{B}'$ has the obvious universal property, i.e. for every homomorphism $f$ from $\mathcal{B}$ to a countably complete Boolean algebra $\mathcal{C}$ there exists a unique homomorphism $g: \mathcal{B}' \to \mathcal{C}$ such that $g \circ i = f$ (it would be nice if $g$ turned out to commute with countable sups too). If no such $\mathcal{B}'$ exists, is there some other useful definition of "smallest" countably complete Boolean algebra containing $\mathcal{B}$?

If it makes any difference, I'm mostly interested in the special case where $\mathcal{B}$ is a direct limit of a sequence of finite Boolean algebras.

Edit: Thanks very much for the replies, it's a shame I can only mark one as the answer. It will take me a while to absorb the various references I've been given, so if I run into difficulty I'll bump the thread with an edit.

Edit 2: Bumping with followup question, please see my answer below.

Answer 1

The short answer is "yes", and it's a special case of a much, much more general theorem on relatively free algebraic constructions.

In other language, you are asking whether the underlying functor from countably complete Boolean algebras to Boolean algebras has a left adjoint. The more general question is whether, given a homomorphism $\phi: S \to T$ between two monads on $Set$, the evident underlying functor

$$Set^\phi: Set^T \to Set^S$$

from the category of $T$-algebras to the category of $S$-algebras has a left adjoint. For this I'll direct your attention to this nLab article.

Of course, we have to know that countably complete Boolean algebras can be described as algebras of a monad on $Set$, but this too follows from general theory. I'll refer you to another nLab article for this; the article is not complete but it should give the idea. The upshot is that for any algebraic theory with only a small set of operations of each arity, there is a corresponding monad on $Set$ whose algebras are the models of the theory. The general constructions go back to work in the sixties, due to Lawvere, Linton, and others.

Edit: I'll remark that had you said "complete" instead of "countably complete", then the answer would have been no. In fact, the underlying functor from complete Boolean algebras to sets has no left adjoint; this is mentioned for instance in Categories for the Working Mathematician. But in your case, the theory is generated by a set of operations and equations, and all is well.

Answer 2

Every Boolean algebra $\mathbb{B}$ embeds densely in its completion as a Boolean algebra, which is a complete Boolean algebra (more than just countably complete). The completion $\bar{\mathbb{B}}$ can be constructed as the regular open algebra, the set of all regular open subsets of $\mathbb{B}-\{0\}$, where the topology is generated by the basic open sets consisting of lower cones. A set is regular open if it is the interior of its closure.

Indeed, the completion operation is extremely general, and is used pervasively in set theory in the context of the forcing technique. Every separative partial order $\mathbb{P}$, and this includes any Boolean algebra (minus $0$), embeds densely in its regular open algebra, and this is always a complete Boolean algebra. This fact is the main connection between the poset-based account of forcing and the Boolean-algebra based account of forcing.

The wikipedia link above lists several universal properties of this completion.

Answer 3

In the course of a forthcoming research project with Asgar Jamneshan (in ergodic theory), we managed to discover a rather explicit answer to this question, which I am recording here if anyone is interested.

Let ${\mathcal B}$ be a Boolean algebra, let $\mathrm{Stone}({\mathcal B})$ be its Stone space (the space of Boolean homomorphisms $\alpha$ from ${\mathcal B}$ to $\{0,1\}$), let $\Sigma$ be the Baire $\sigma$-algebra of $\mathrm{Stone}({\mathcal B})$. Then there is a natural inclusion $\iota \colon {\mathcal B} \to \Sigma$ defined by sending each $E \in {\mathcal B}$ to the clopen set $\{ \alpha \in \mathrm{Stone}({\mathcal B}): \alpha(E) = 1 \}$. This is a Boolean homomorphism which is universal in the sense that for every Boolean homomorphism $f \colon {\mathcal B} \to {\mathcal X}$ into a $\sigma$-complete Boolean algebra ${\mathcal X}$ there is a unique lift $\tilde f: \Sigma \to {\mathcal X}$ such that $f = \tilde f \circ \iota$.

• Proof of uniqueness: the clopen sets in the Stone space $\mathrm{Stone}({\mathcal B})$ separate points, hence by Stone-Weierstrass the Baire algebra is generated by the clopen sets, hence $\Sigma$ is generated by $\iota({\mathcal B})$, giving uniqueness.
• Proof of existence: Stone duality gives a continuous map from $\mathrm{Stone}({\mathcal X})$ to $\mathrm{Stone}({\mathcal B})$, which pulls back Baire sets to Baire sets, thus maps $\Sigma$ to Baire sets in $\mathrm{Stone}({\mathcal X})$, each one of which can be associated to an element of ${\mathcal X}$ by the Loomis-Sikorski theorem (every Baire set in $\mathrm{Stone}({\mathcal X})$ is equivalent up to meager sets to a unique clopen set). One easily verifies that this gives a map $\tilde f: \Sigma \to {\mathcal X}$ with the required properties.

[EDIT: a previous version erronously assumed that the pullback of a meager set was meager, leading to an incorrect conclusion. Now corrected.]

Answer 4

You can also look at III.3.11 in Johnstone's book on Stone Spaces:

http://books.google.com/books?id=CiWwoLNbpykC&lpg=PP1&dq=stone%20spaces&pg=PA108

Answer 5

This question was answered in topological terms by J. Vermeer in The smallest basically disconnected preimage of a space. Topology Appl. 17 (1984), no. 3, 217–232. See here for a review and here for the paper.

Answer 6

Hi all,

I've just come up with something that's relevant to the question I asked here. I'm bumping the thread partly in case anyone else cares, and partly in case (as is more likely) I've made an error and someone can point it out. Anyway: I believe I can prove that the $\sigma$-algebra generated by a Boolean algebra $A$ (in the sense of Todd's answer, i.e. the image of a left adjoint to the forgetful functor from $\sigma$-algebras to Boolean algebras) has a rather natural representation, namely as the $\sigma$-field generated by the double dual of $A$, i.e. the smallest $\sigma$-field containing all the clopen subsets of the dual space of $A$. Here is the proof:

Let $A$ be a Boolean algebra, let $A^\star$ be its dual Boolean space and $A^{\star \star}$ the dual algebra of its dual space, i.e. the set of clopen subsets of $A^\star$. Let $\bar{A}$ be the $\sigma$-algebra of Baire sets in $A^\star$, i.e. the $\sigma$-field of subsets of $A^\star$ generated by $A^{\star \star}$. Let $\alpha: A \cong A^{\star \star}$ be the canonical isomorphism, and let $\eta: A \to \bar{A}$ be the composition of $\alpha$ with the inclusion.

Suppose given a $\sigma$-algebra $B$ and a homomorphism (of Boolean algebras) $h: A \to B$. Define $B^\star$, $B^{\star \star}$, $\bar{B}$ and $\beta: B \cong B^{\star \star}$ as before. By Theorem 41, p. 376 of [1], $B^\star$ is a $\sigma$-space, i.e. the closure of every open Baire set is open. By Theorem 42, p. 381, there is a $\sigma$-homomorphism $\phi: \bar{B} \to B^{\star \star}$ such that $\phi$ maps ever clopen set to itself.

$\beta h \alpha^{-1}$ is a homomorphism $A^{\star \star} \to B^{\star \star}$, so by duality there is a unique continuous function $f: B^\star \to A^\star$ such that $f^{-1} P = \beta h \alpha^{-1} (P)$ for every $P \in A^{\star \star}$. It is easy to see that $f^{-1} S$ is a Baire set whenever $S$ is, so define

$f^\star : \bar{A} \to \bar{B}$; $S \mapsto f^{-1} S$.

$f^\star$ is clearly a $\sigma$-homomorphism. Let

$\bar{h} \equiv \beta^{-1} \phi f^\star: \bar{A} \to B$.

Then one may check, using the defining property of $f$ and the fact that $\phi$ maps clopen sets to themselves, that $\bar{h} \eta = h$. The uniqueness of $\bar{h}$ with this property follows from the fact that the range of $\eta$ generates $\bar{A}$.

So there's the alleged proof; I can't see anything wrong with it but the result strikes me as being "too good to be true", and if it is true then I'm surprised I didn't see any reference to it online before I started this thread. So I'll be grateful if anyone can spot a mistake.

[1] Steven Givant and Paul Halmos, Introduction to Boolean Algebras, Springer 2009