Talk:Stone's representation theorem for Boolean algebras
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Talk moved from former "Stone's duality". --Markus Krötzsch 14:39, 18 Apr 2004 (UTC)
- Stone's duality generalises to infinite sets of propositions the use of truth tables to characterise elements of finite Boolean algebras.
I think of a truth table as a device to define logical operators, i.e. to define functions {T,F}n→{T,F}, and of the elements of a Boolean algebras as logical propositions that can be combined with the operators and, or, not. So I don't quite understand how truth tables are used to characterise elements of finite Boolean algebras. Could that be explained a bit more?
Also, a prominent Stone space is the Cantor set; does it correspond to an interesting Boolean algebra? AxelBoldt 16:06, 13 Feb 2004 (UTC)
Well, to a whole bunch of them. The homeomorphism class of the Cantor set includes many spaces that come up (p-adic integers, typical profinite Galois groups, etc., etc.). Typically the clopen sets are something relatively easy to describe in such examples, and so you get a Boolean algebra that makes some sense. But what makes any one 'presentation' of what is the same Boolean algebra (up to iso) 'interesting'? It's the kind of thing that any two different people might answer two different ways.
Charles Matthews 16:14, 13 Feb 2004 (UTC)
I guess another way of putting roughly the same point is that the self-homeomorphism group of the Cantor set is very large - much bigger than the group of homeomorphisms that preserve the order of reals. The latter is already large, but not impossible to picture. The former is really hard to think about (logicians are welcome to it, in my view). But it clearly respects clopen sets, so gives one a huge way of talking about 'the' Boolean algebra.
Charles Matthews 16:26, 13 Feb 2004 (UTC)
Move
editWhy was this article moved from Stone's duality, when that is a term much more likely to be used for the subject of this article (and Stone duality more so)? --- Charles Stewart 19:51, 9 Jun 2005 (UTC)
Wouldn't Stone space be the natural home for this material? Moving it to Stone duality would be like moving group (mathematics) to symmetry (mathematics). Symmetry (mathematics) should simply be a redirect to group (mathematics). --Vaughan Pratt 13:52, 20 August 2007 (UTC)
On second thoughts the current redirect for Symmetry (mathematics), namely to Symmetry in mathematics, is better since group associativity is on the nose whereas that of symmetry in general need not be, e.g. the symmetry of cartesian product, tensor product, etc. --Vaughan Pratt 16:26, 21 August 2007 (UTC)
Simpler statements of the theorem
editThe simplicity of the properly stated theorem and its elegant consequences are over shadowed by the interjection of categorical language. Also, at what point in the history of Stone's representation theorem, has category theory had any place? -- Differentiablef — Preceding unsigned comment added by Differentiablef (talk • contribs) 06:57, 20 January 2011 (UTC)
Section 1 includes a more traditional and elementary statement of the theorem, which I picked up from Stoll (1963). Those of you with serious training in Boolean algebra should feel free to edit it.
I see that the Wikipedia philosophy community is watching this entry. I have no objections to that fact, but wish to point out that the implications of the Stone theorem for philosophy and logic are less than evident.
This entry should reference a proof that any college senior specializing in algebra should be able to follow givewn close reading. Other than Stoll (1963), I suggest Halmos and Givant (1998: 76-77).132.181.160.42 22:39, 23 September 2007 (UTC)
First Example?
editIt's quite a statement to say that the Stone duality was the 'first' example of a non-trivial duality of categories - there are plenty of other great classical theorems far predating it which may also be phrased as giving (non-trivial) dualities of categories. Unless there is a qualification given to this, I'm taking it out. —Preceding unsigned comment added by 137.158.152.207 (talk) 07:11, 11 November 2010 (UTC)
Strange phrase
edit"Note that B embeds in S(B) by sending each element b to the principal ultrafilter generated by b." — What could it mean?? Elements of B are, basically, clopen subsets of S(B), not points of S(B). Boris Tsirelson (talk) 17:47, 1 September 2011 (UTC)
- Well, this unfortunate sentence means "B can be embedded into S(B)" (by some injective mapping defined immediately after), rather than "B is a subset of S(B)". What do you suggest? Would it be better to write: "Note that B can be embedded..."? Kope (talk) 20:39, 9 September 2011 (UTC)
- No, I wonder about the essence of that claim, not just the form of the phrase. Let us take the simplest case: a finite Boolean algebra. It contains 2n elements, where n is the number of points in S(B). In which sense can the 2n-element set be embedded into the n-element set? Boris Tsirelson (talk) 16:40, 10 September 2011 (UTC)
- The other way around.An n-element set into a 2n-element set. Kope (talk) 13:59, 11 September 2011 (UTC)
- That is, "S(B) can be embedded into B"? First, this is not the claim in the article. Second (and worse), it works for the simplest case (of finite Boolean algebras), but fails in general. Take for instance an atomless countable Boolean algebra (it is unique up to isomorphism). Its Stone space is the Cantor set (up to homeomorphism). Which way do you embed the continuum of points of the Cantor set into the countable Boolean algebra? Boris Tsirelson (talk) 19:11, 11 September 2011 (UTC)
- Nope. B has n elements and it embeds in the 2n-element S(B). Kope (talk) 21:19, 11 September 2011 (UTC)
- That is, "S(B) can be embedded into B"? First, this is not the claim in the article. Second (and worse), it works for the simplest case (of finite Boolean algebras), but fails in general. Take for instance an atomless countable Boolean algebra (it is unique up to isomorphism). Its Stone space is the Cantor set (up to homeomorphism). Which way do you embed the continuum of points of the Cantor set into the countable Boolean algebra? Boris Tsirelson (talk) 19:11, 11 September 2011 (UTC)
- The other way around.An n-element set into a 2n-element set. Kope (talk) 13:59, 11 September 2011 (UTC)
- No, I wonder about the essence of that claim, not just the form of the phrase. Let us take the simplest case: a finite Boolean algebra. It contains 2n elements, where n is the number of points in S(B). In which sense can the 2n-element set be embedded into the n-element set? Boris Tsirelson (talk) 16:40, 10 September 2011 (UTC)
Perhaps what they mean is that $B$ can be embedded into the powerset algebra of $S(B)$. The text in the article would make sense if they were thinking of $S(B)$ as that algebra rather than as a topological space. Of course then the sentence should have said "to the principal filter generated by b", because e.g. 1 does not generate a principal ultrafilter. Given that problem, I went ahead and removed the sentence. — Carl (CBM · talk) 11:25, 12 September 2011 (UTC)
- And then it should be also explained that a principal filter is related to a subset of the Stone space... Boris Tsirelson (talk) 11:32, 12 September 2011 (UTC)
- We crossed edits. I added a sentence just now about how the isomorphism is defined, is that what you were thinking of? — Carl (CBM · talk) 11:34, 12 September 2011 (UTC)
Singleton or element
editIt says on this page {x \in S(B) : b \in x } Where b is an element of $B$ When defining the basis. If x \in S(B), then it is an ultrafilter on B, meaning it should be a set of subsets of B. Thus, b \in x notationally implies b is a subset of B rather than an element of B. — Preceding unsigned comment added by 131.215.220.161 (talk) 21:34, 14 April 2016 (UTC)