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Someone has asked a question at BoolALg without giving a name. I have suggested they re-ask that here (and say who they are).
Yes, the questioner is right. Here is a useful way of thinking about it: Boolean algebras are equivalent to Boolean rings (where binary addition corresponds to symmetric difference), and so we need to form the free Boolean ring. This is analogous to polynomial algebras, where one forms the free vector space generated by monomials (which form a free monoid or free commutative monoid).
Thus, the right recipe is to take the free $\mathbb{Z}_2$-vector space generated by the commutative monoid of idempotent monomials. An idempotent monomial can in turn be regarded as a finite subset of the alphabet. So, the free Boolean algebra on $X$ consists of functions $P_{fin}(X) \to \mathbb{Z}_2$ of finite support.
I’ll make the necessary changes.
From the timing can we tell if it’s likely the questioner was brought here by the Cafe post? It would be good to know it’s possible to bring in some external scrutiny.
I have carried out some revisions at BoolAlg. I left a little for Toby at the end to fix up as regards the constructive aspects, if he’d like to have a go.
I half-wonder whether the questioner is Qiaochu Yuan, a prolific patron at Math Overflow, who recently wrote a post on the Stone representation theorem at his blog Annoying Precision. He’s presently at Cambridge University.
Anyway, please have a look.
Toby, it looks as though you just restored what was there earlier, that questioner had a problem with.
I hope that I didn’t restore anything! I just added a header above questionable material that you left. I still need to rewrite that section (or determine that it’s not necessary.)
@ Todd
If you’re still working on it, I can cancel my edit for now and work on it off line.
The visitor was G. Rodrigues. He came back after I added my query box.
Oh, sorry Toby. I’m done for now, if you want to work on it (at your leisure).
OK, I think that it’s good now.
Also, thanks to Rodrigues and Todd for fixing this page up; I don’t know why I wrote that wrong stuff before!
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