Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexSomeone has asked a question at [[BoolALg]] without giving a name. I have suggested they re-ask that here (and say who they are).

   Someone has asked a question at BoolALg without giving a name. I have suggested they re-ask that here (and say who they are).

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexYes, 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexFrom 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexToby, it looks as though you just restored what was there earlier, that questioner had a problem with.

   Toby, it looks as though you just restored what was there earlier, that questioner had a problem with.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI 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.)

   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.)

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex@ Todd If you're still working on it, I can cancel my edit for now and work on it off line.

   @ Todd

   If you’re still working on it, I can cancel my edit for now and work on it off line.

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThe visitor was G. Rodrigues. He came back after I added my query box.

   The visitor was G. Rodrigues. He came back after I added my query box.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOh, sorry Toby. I'm done for now, if you want to work on it (at your leisure).

   Oh, sorry Toby. I’m done for now, if you want to work on it (at your leisure).

10. • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeNov 26th 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexOK, I think that it's good now.

    OK, I think that it’s good now.

11. • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeNov 26th 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexAlso, thanks to Rodrigues and Todd for fixing this page up; I don't know why I wrote that wrong stuff before!

    Also, thanks to Rodrigues and Todd for fixing this page up; I don’t know why I wrote that wrong stuff before!